ELEMENTARY 



LESSONS IN PHYSICS 



MECHANICS {INCLUDING HYDROSTATICS) 
AND LIGHT 



BV . 

EDWIN H. HALL, Ph.D. 

Assistant Professor of Physics in Harvard College 




NEW YORK 
HENRY HOLT AND COMPANY 

1900 



TWO COPIES RECEIVED, 

Library of Congre«% 
Office of the 

FEB 7 - 1900 

Ktglttar of Copyright* 






54196 



Copyright, 1900 

BY 

Henry Holt & Co t 



SECOND COPY, 



^ O O (o 



ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. 



INTRODUCTION. 

This volume, which is the First Part of Hall and Ber- 
gen's revised Text-booh of Physics (1897), may be regarded 
as the second edition of Hair's Elementary Lessons in 
Physics. It is intended for the use of pupils in the early 
years of a high-school course or even the last year of a 
grammar-school course, and it assumes no previous system- 
atic study of physics. 

The course of study here given includes laboratory work, 
to be done by the pupils, combined with a considerable 
amount of general 'instruction, to be illustrated by lec- 
tures given by the teacher. The laboratory work is mainly 
or wholly quantitative, as it must be for large classes, quali- 
tative laboratory work in such classes making impossible 
demands upon the time and energy of the teacher. 

A First Course made up, as this one is, of simple experi- 
ments in mechanics (including hydrostatics) and optics, 
more difficult matters in mechanics, together with heat, 
sound, electricity, and magnetism, being deferred, is un- 
usual; but it is here proposed as better suited to many pu- 
pils and to many schools than the more familiar practice of 
going through the whole of mechanics before entering upon 
any other part of physics, and putting light, or optics, after 
heat. The laboratory outfit required for these early exer- 
cises is much less complicated and expensive than that 
required for much of the later laboratory work ; so that 
many schools which would be quite unable to offer a labo- 

iii 



iv INTRODUCTION. 

ratory course extending over the whole field of elementary- 
physics will find it possible to do what this book requires. 
Moreover, those teachers w r ho have ample means and facil- 
ities, and who intend to take their pupils through the 
whole range of elementary laboratory work, will find it 
advantageous to interrupt the course in mechanics lest their 
classes grow tired of what is, for many young pupils, the 
least interesting part of the study. 

The book follows, as a rule, the method of leading up to 
the statement of laws by means of carefully chosen experi- 
ments, rather than the opposite one of giving experiments 
as illustrations or proofs of laws already stated. It can 
hardly be said for the former method that it teaches the 
art of making discoveries, — that art is as difficult to teach 
as the art of getting rich, — but it has a tendency to keep 
the pupil in a more active, self-dependent state of mind 
than the latter method, and in particular it prevents iu a 
large measure that state of bias, or preconception, in the 
performance of experiments, which is so dangerous not 
merely to accuracy of observation but to mental rectitude. 
On the other hand, the teacher using the method of this 
book must not allow his pupils to think that their experi- 
ments, even when most satisfactory, really demonstrate the 
rigid accuracy of any numerical law, — the law of a balanced 
lever, for instance. He should ask of them, " What law do 
your experiments indicate as true ?" and after their answer 
he should tell them whether their inference is or is not in 
accordance with the opinion held by those best qualified 
to judge of the matter in question. 

It is the firm conviction of the writer that class labora- 
tory work not accompanied by persistent, energetic teach- 
ing is sure to be a failure. We are often told that the 
favorite method of the elder Agassiz with a new pupil was 



INTROD UCTION. V 

to set him to gaze in solitude at a single fish for two or 
three days. Those who would make this the model for 
science-teaching in general forget that pure observation 
of numerous, minute, varied details plays a much more 
important part in natural history than in physics. The 
teacher of physics who would produce good and lasting 
results must see to it not merely that the laboratory work 
shall be carefully done, but that the proper lessons shall 
be drawn from it and the proper applications made. In 
fact, the young pupil should give as much time to the 
study of physics in the lecture- or recitation-room as in the 
laboratory proper. 

The course of study described in this book is intended 
to run through a school year and to occupy the pupil at 
least two school-periods, each forty minutes long, or more, 
per week; one usually in the laboratory, and the other in 
the lecture- or recitation-room. The number of laboratory 
Exercises is considerably less than the number of school- 
weeks in the year, but some of them may prove to be too 
long for a single school-period, and teachers will welcome 
an occasional opportunity for repetition or review. The 
amount of time required for the course will depend some- 
what upon the age of the pupils taking it, and classes in 
the first year of a high-school course may find three 
school-periods a week for one year none too much time for 
doing the work well. 

It is highly desirable that pupils whose laboratory work 
is confined to that described in this book should have lec- 
ture-room illustrations of many things not here dealt with, 
elementary facts and principles in heat, sound, electricity, 
and magnetism. It is therefore recommended that every 
teacher of such pupils be provided with the means neces- 
sary for such illustrations, for example much of the appa- 



vi INTRODUCTION. 

rat us designated by Roman numerals in Hall and Bergen's 
Physics. 

The following estimates of cost for apparatus and ma- 
terials are only approximate. It is hardly possible to make 
an accurate estimate, as prices will vary from time to time, 
and different dealers have somewhat different grades of ap- 
paratus. The cheapest is not necessarily the best to buy. 

FOR THIS BOOK. 

Teacher's apparatus and supplies, pp. 174-178 .... $78.00 
Students' apparatus, pp. 170-173, for each member 

of a laboratory squad 6.00 

Table, accommodating six workers, p. 178 25.00 

Total for all Exercises and Experiments of this 

book, with laboratory squads limited to twelve.. . 200.00 

FOR THE SECOND PART OF HALL AND BERGEN'S 

PHYSICS. 

Teacher's apparatus, pp. 581-586 350.00 

By omittiug the thermopile and accompanying 
apparatus and the Roentgen-ray apparatus this ex- 
pense can be reduced about $100. 

Apparatus for the course can be obtained from the fol- 
lowing well-known manufacturers: 

The Chicago Laboratory Supply and Scale Company, 
39 West Randolph Street, Chicago. 

The Franklin Educational Company, Harcourt Street, 
Boston. 

The Knott Company, 16 Ashburton Place, Boston. 

The Ritchie Company, Brookline, Mass. 

The Ziegler Electric Company, 141 Franklin Street, 
Boston. 

E. II. H. 

January 16, 1900 



TABLE OF CONTENTS. 



PAGE 

Introduction iii 

CHAPTER I. 

INTRODUCTORY. 

Definition of Physics. — Use of Physics. — Qualitative and 
Quantitative Knowledge. — Object of this Course.— Preliminary 
Exercises: Measurement of Distance, of Area, of Volume, — with 
Estimation of Errors 1 

MECHANICS. 

CHAPTER II. 

DENSITY AND SPECIFIC GRAVITY. 

Density.— EXERCISE 1: Weight of Unit Volume of a Sub- 
stance.— Density of Water.— Weight.— Mass.— EXERCISE 2: 
Lifting Effect of Water.— EXERCISE 3: Specific Gravity of a 
Solid that will Sink in Water.— EXERCISE 4: Specific Gravity of 
Wood by Use of Sinker.— EXERCISE 5: Weight of Water Dis- 
placed by a Floating Body. — EXERCISE 6: Specific Gravity by 
Floating Method.— EXERCISE 7: Specific Gravity of a Liquid by 
two Methods 15 

CHAPTER III. 

FLUID-PRESSURE. 

Fluids: Liquids and Gases— Experiments with Pressure- 
gauge in Water. — Torricelli's Experiment. — Atmospheric Pres- 

vii 



viii TABLE OF CONTENTS. 

PAGE 

sure. —Barometer. — Boyle's Law.— Hydraulic Press. — Water- 
pumps. — Siphon. — Balancing Columns 28 

CHAPTER IV. 

THE LEVER, 

EXERCISE 8 : The Straight Lever, First Class, — Circular 
Lever.— EXERCISE 9: Centre of Gravity and Weight of Lever. — 
EXERCISE 10: Levers of the Second and Third Class: s.— EXER- 
CISE 11: Force Exerted at the Fulcrum. — Pulleys. — General Law 
for Relation of Power to Weight 41 

CHAPTER Y. 

THREE FORCES ACTING THROUGH ONE POINT — THE PARALLELO- 
GRAM OF FORCES. 

Introductory. — EXERCISE 12: Errors of a Spring-balance. — 
EXERCISE 13: Parallelogram of Forces.— The Inclined Plane: 
Wedge, Screw. — Equilib i ant aDd Resultant 61 

CHAPTER VI. 

FRICTION. 

EXERCISE 14: Friction between Solid Bodies.— EXERCISE 15: 
Coefficient of Friction.— Friction in Applied Mechanics.— Rolling 
Friction. — Friction between Solids and Fluids , 78 

CHAPTER VII. 

THE PENDULUM. 

Use in Clocks.— Experiments.— Springs in Place of Pendu- 
lums 86 

LIGHT. 

CHAPTER VIII. 

Nature of Light— Visibility of Objects. 

Light is Something that Travels. — Velocity.— A Wave- 
motion.— Pencils and Rays.— Shadows.— EXERCISE 16: Use of 



TABLE OF CONTEXTS. ix 

PAGE 

Rumford Photometer. — Bunsen's Photometer. — Effect of Body 
on which Light Falls. — Visibility of Objects 90 

CHAPTER IX. 

REGULAR REFLECTION OF LIGHT. 

EXERCISE 17: Images in a Plane Mirror. — Images of Images, 
Kaleidoscope. — EXERCISE 18: Images Formed by a Convex 
Cylindrical Mirror. — EXERCISE 19 : Images Formed by a Concave 
Cylindrical Mirror. — Relation of Cylindrical to Spherical Mir- 
rors. — The Ophthalmoscope. — Formulas relating to Curved 
Mirrors 103 

CHAPTER X. 

REFRACTION OF LIGHT. 

Introductory Experiments. — Angles of Incidence and Re- 
fraction.— EXERCISE 20: Index of Refraction of Glass.— EX- 
ERCISE 21 : Index of Refraction of Water.— Index Different for 
Different Colors. — Index of Refraction and Velocity. — Internal 
Reflection, Critical Angle. — Transparent Plates and Prisms. — 
Dispersion, the Spectrum.— Lenses.— EXERCISE 22; Focal 
Length of Converging Lens.— EXERCISE 23 : Conjugate Foci of 
a Lens. — EXERCISE 24 : Shape and Size of Real Image Formed by 
a Lens. — EXERCISE 25 : Virtual Image Formed by a Lens. — 
Spherical and Chromatic Aberration in Lenses. — Achromatic 
Lenses. ..,,,,,,,, 123 

CHAPTER XL 

THE EYE : SIGHT AND COLOR. 

Description of the Eye.— Accommodation, etc. — The Color- 
sense. — Mixing Color Impressions.;— Complementary Colors. — 
Fatigue of Retina. — After-images 152 

CHAPTER XII. 

OPTICAL INSTRUMENTS. 

Photographer's Camera. — Magic Lantern. — Projecting a Spec- 
trum. — Simple Microscope. — Compound Micro-cope. — Tele- 
scope 156 



X TABLE OF CONTENTS. 

APPENDIX I. 

PAGE 

Focal Length, etc., of Lenses and Combinations 168 

APPENDIX II. 
Indices of Refraction e . 169 

APPENDIX III. 

List of Apparatus, etc., for the Exercises and Experiments of 
the Preceding Pages 170 



ELEMENTS OF PHYSICS. 

CHAPTER I. 
INTRODUCTORY. 

1. Definition of Physics. — Physics is the science of. 
mechanics, heat, sound, light, electricity, and magnetism,. 
Everybody knows something about these things before he 
begins to study them in a regular way, but sometimes he 
does not know them by the names which are given to them 
in books. 

2. Use of Physics. — In sailing boats or flying kites, in 
walking or swimming, in almost any kind of bodily work or 
play, we have to do with physics, that part of physics which 
is called mechanics. We learn to do many mechanical acts 
very well indeed by observation and experience, without 
thinking very much about them or knowing exactly how we 
do them ; but when w T e have to do something that we have 
never done before and have never seen any one else do, 
something, perhaps, that nobody ever did before, we must 
think and study. 

3. Illustrations. — Thus no man who has practiced swim- 
ming need study mechanics to improve himself in that art; 
but if he would build a ship and make it swim through all 
kinds of weather and water, he must study mechanics a 
good deal in order to know what size and shape to give the 
various parts, how best to put them together, and how to 



2 PHYSICS. 

balance the whole. If it is to be a steamship, some one 
must know a good deal about heat, to make the furnaces 
and boilers right. We must know about magnetism to 
make and use the ship's compass. We may use electricity 
to furnish light on board at night. We must study sound 
in order to make the best fog-signals to guard against colli- 
sions and shipwreck in thick weather. 

In short, the science of physics in all its main divisions 
is not only a very interesting study to many minds, but it 
is of great use to civilized mankind. Man has become 
civilized, indeed, not by merely imitating what his fathers 
have done, but by studying, that is, observing and thinking, 
and gradually improving upon the work of those who have 
gone before him. 

4. Qualitative Knowledge. — Everybody knows that a 
piece of wood will float in water, and that a stone will sink. 
Everybody knows that if a stick and a stone are tied 
together and put into water, the stone tends to sink the 
stick, and the stick tends to float the stone. This kind of 
knowledge is called qualitative. It tells in a general way 
how the stick and the stone act toward each other. 

5. Quantitative Knowledge. — Some people know enough 
about the laws of flotation to calculate with accuracy how 
large a stick of a known kind of wood will be needed to 
float a stone of known size and weight. They have what 
is called quantitative knowledge of the matter. They can 
tell how much the stone will pull down on the wood, and 
the wood pull up on the stone, when the two are together 
in water. 

Everybody knows that a beam has a quality which we call 
strength — it can bear a load. This is qualitative knowl- 
edge. Everybody knows that a thick beam is stronger than 
a thin beam. This is quantitative knowledge of a kind, a 
rather indefinite kind. Some people know how much 



INTROD UCTOR Y. 3 

stronger the thick beam is than the thin beam. They have 
a more complete quantitative knowledge. 

6. Comparison of the Two Kinds of Knowledge. — It is 

evident that quantitative knowledge is more useful than 
mere qualitative knowledge. The former includes the latter. 
The qualitative man says, " I want to build a house. I 
shall need some land to put it on, and some beams and 
boards and bricks," etc. The quantitative man says, 
" Yes, you will need ail these things, but if you don't make 
your ideas more precise you will have a lumber-yard when 
you have done, not a house." 

7. Object of this Course in Physics. — The knowledge of 
physics which children and older people get by merely 
knocking about in the world is mostly qualitative. The 
course of work laid out in this book is intended to add greatly 
to the pupil's stock of this kind of knowledge, and to do 
something more. It aims to make the pupil familiar with 
quantitative work, and to give him a considerable amount 
of quantitative knowledge. 

We shall begin at once with simple introductory measure- 
ments. All the exercises of this chapter are called Prelim- 
inary Exercises. 

Measurement of Distance. 

8. The Straight Line. — The line to be measured may be 
along the edge of a table (or sheet of paper) from one fine 
scratch to another, a distance of about 15 inches. It is a 
great convenience to have all the pupils 
measure equal distances; accordingly, 
the teacher is advised to lay off these 
distances by some method like the fol- 
lowing: A carpenter's square is placed a~ 
along the edge of the table as in Fig. 1, FlG - 1 - 

and while it is held firmly in place a fine light scratch is 



4 PHYSICS. 

made with the point of a sharp knife-blade at right angles 
with the edge of the table at the points a and b. The dis- 
tance from a to i is the one to be measured by the pupil. 
The first-described method of using the measuring-stick in 
the following Exercise is not a good method, but it is one 
that many will use if they are not properly instructed. 
The second method is a good one, and the two are here 
brought together in order that the pupil may see at once 
the right way and the wrong way to use such an instrument. 
Much of the interest and profit of this Exercise will come 
from the opportunity given each pupil to compare his own 
work with that of others. 

EXERCISE A. 

MEASUREMENT OF A STRAIGHT LINE. 

Apparatus : A short measuring-stick (No. 1) * and a meter-rod 
(No. 2). 

To each pupil is given a measuring-stick about one-fourth as long 
as the distance from a to b. We will suppose that these sticks are 
made by sawing a meter-rod, graduated to millimeters, into ten 
equal parts. The saw-cut will usually leave the divisions at the 
very ends of the sticks imperfect, and these divisions should not be 
used in the measurements. 

Let each pupil measure his distance at least twice carefully, with 
his measuring-stick laid flat upon the table, the marks upon the 
stick being thus horizontal, and let him write upon the blackboard 
the results of his two measurements. 

Then let each pupil measure his distance twice again, this time 
placing his measuring-stick upon its edge, so that the marks upon it 
will be vertical, making a light, fine mark upon the table with a 
sharp pencil to set the stick by, whenever it is moved forward a 
length. These new measurements are also to be placed upon the 
blackboard under the first ones. 

Finally let each pupil measure his whole distance at once with his 
meter-rod and write this last measurement with the others. 

* Any piece of apparatus to be used in the Exercises will usually 
be referred to by the number it bears in the list of apparatus given 
at the end of the book. 



INTBODUCTOBT. 5 

9. Errors. — To judge of the accuracy of a set of measure- 
ments it is not enough to know how much these differ 
among themselves, for the importance of the difference 
usually depends upon the ratio which the difference bears 
to the whole quantity measured. A thousandth part of an 
inch might be a very serious difference to a watchmaker in 
the measurement of some small cylinder, while a difference 
of several inches in the measurement from one mile-post to 
another would be of little consequence. The pupil should 
therefore form the habit of comparing his errors, or the 
differences of his measurements, with the whole quantity 
that he had to measure. 

Let us suppose, for instance, that in Exercise A the 
measurements made by one pupil are 37.30 cm., 37.00 cm., 
and 37.10 cm. The greatest difference is found between 
the first and second. It is 0.3 cm., and its ratio to 37.15 
cm., which is midway between 37.30 cm. and 37.00 cm., 
is 0.0081 — . We see, then, that the difference between the 
two measurements of the line is about eight one-thousandths, 
not quite one per cent, of the length of the line. 

Each pupil should make a similar calculation from his 
own measurements in Exercise A. 

10. Units and Standards of Measurement. — The impor- 
tance of having definite units of length, of weight, etc., so 
that any man in dealing with his neighbor may know just 
how much is meant by the words foot, pound, and the like, 
is so great that in all civilized countries the exact meaning 
of such words is fixed by law, and very great care is taken 
to make and preserve government standards, as they are 
called, standard yard-sticks, standard pound-weights, for 
instance, with which as patterns the measuring instruments 
used in business are compared and tested. 

Interesting accounts of the foot, the yard, the meter, etc., 
can be found in almost any encyclopedia. 



6 PHYSICS. 

Meter-rods for school use are in many cases marked off 
in inches on one side. With the information given by such 
a rod, the class can find how many centimeters are equal to 
one inch. This number carried to two places of decimals 
is accurate enough for most purposes. 

11. The Right Triangle. — Eight triangles, that is, tri- 
angles having one right angle (see Fig. 2), are much used in 
the study and application of physics. In such triangles 




Fig. 2. 

there is a simple and important relation between the length 
of the longest side and the length of the other two sides. 
Part 1 of the following Exercise B is intended to show this 
relation and at the same time to give practice in measure- 
ment. 

12. Circles. — The relation between the length of the 
diameter of a circle and the length of its circumference is 
also very frequently used in physics. Part 2 of Exercise B 
has to do with this relation. 



EXERCISE B. 

THE LINES OF THE RIGHT TRIANGLE AND THE CIRCLE. 

Apparatus : A 30-cm. measuring- stick (No. 3). A sheet of paper 
upon which is drawn carefully a right triangle no side of which is 
less than 10 cm. long. (No two pupils should use exactly similar 



INTRODUCTORY. 7 

triangles.) A cylinder of wood 4 or 5 cm. in diameter (No. 4). A 
narrow straight-edge strip of thin paper. 

Part 1. Measurement of the Sides of a Right Triangle. 

— Let each pupil measure very carefully all the sides of his triangle, 
not being content to read to the nearest 0.1 cm., but striving to note and 
measure 0.05 cm. distances, if he can do so without hurting his eyes. 

After the measurements are made, square the length of each side 
and compare the greatest square with the sum of the other two 
squares. The conclusion drawn from this comparison must not be 
extended to triangles which are not right-angled. 

Part 2. Measurement of the Circumference and Diam- 
eter of A Circle. — Measure carefully the diameter of one end of 
the cylinder. Then wrap the strip of paper around the curved sur- 
face of the cylinder at the same end, and mark upon the edge of the 
strip the point where the second winding of the paper begins to 
overlap the first. Then unfold the paper and measure upon it that 
distance which extended once around the cylinder. Then divide this 
distance, which of course is equal to the circumference of the circle, 
by the length of the diameter. The ratio thus obtained is one which 
it is important to know, although we shall not have much occasion 
to use it in this book. Mathematicians, physicists, and engineers 
use it so much that thay have a particular sign, it, to denote it. 

This sign is a Greek letter and is called pe by students of Greek, 
but when used as just described it is often called pi to distinguish 
it from p. 

13. Discussion of Exercise B. — The measurements of 
Exercise B may be discussed somewhat as follows: The 
square of the longest side of the triangle is found by one 
pupil to be 404.01, and the sum of the squares of the other 
two sides 406.05. If the two short sides were measured 
correctly, how large an error in the measurement of the 
longest side would cause the disagreement here found ? 
The long side was measured as 20.10 cm. If it had been 
called 20.20 cm., its square would have been 408.04, which 
is about as much too large as the square actually found is 
too small. If the distance had been measured as 20.15 cm., 
the square would have been 406.02, a quantity very close 



8 physics. 

indeed to the sum of the other two squares. If, therefore, 
the original error lay entirely in the measurement of the 
longest side, this error must have been very nearly 0.05 cm. 
Of course the error may have been made in measuring the 
other sides, or in drawing the triangle, or in all parts of the 
work. An error which mistakes 20.15 for 20.10, or 201.5 
for 201.0, or 2015 for 2010, is called in each case an error 
of 5 parts in 2015, or 1 part in 403, or an error of about 
i per cent (see remarks following Exercise A). 

QUESTION. 

In the case of the circle, which would make the greater difference 
in the result (circumference -5- diameter), an error of 0.05 cm. in the 
measurement of the diameter or an error of 0.10 cm. in the measure- 
ment of the circumference ? 

Measurement of Area. 

14. Unit of Area. — Thus far we have been measuring 
lines. To measure a line, as we see, is merely to find out 
by trial that it is so many centimeters or inches long. A 
line 10.6 cm. long is one that could be divided into ten full 
centimeters and six tenths of another centimeter. We here 
call the centimeter our unit of length. 

If we have to measure a surface, the whole table-top, for 
instance, our task is to find the number of square centi- 
meters, or square inches, or square feet, that would be 
required to cover it, or that it would make if it were cut 
up without waste into squares. In this case the square 
centimeter, or square inch, or whatever square we choose 
to take, is the unit of area. We might set about to measure 
surfaces by actually placing a little square, a square centi- 
meter, for instance, on the given surface, marking a line* 
close around it, then moving it to a new place, marking 
around it, and so on till we had marked off the whole sur- 
face \n\o little squares, with perhaps some fractions of 



INTRODUCTORY. 



9 



squares. But this is not the common or the best way of 
measuring surfaces. The common way is to measure the 
length of certain lines on the surface and from the lengths 
of these lines to calculate the extent of the surface. 



15. Measurement of Rectangles. — If the surface is in the 
form of a rectangle, like Fig. 3, it is plain 
that we have merely to multiply the 
number of units, centimeters let us say, 
in the length by the number of centime- 
ters in the width, and the result, 8x1 
= 32 in this figure, is the number of 
square centimeters into which the surface can be divided. 
This is called the extent or area of the surface. 

In the next Exercise we shall undertake to find rules for 
the measurement of surfaces not quite so simple in shape 
as the rectangle shown in Fig. 3. These will be of the 
class called parallelograms. 



Fig. 3. 



16. Parallelograms. — A parallelogram is a flat figure 
bounded by four straight linos, each line being parallel to 
the line opposite. Thus A and B in Fig. 4 are parallelo- 




B 



Fig. 4. 

grams. A is what we have just called a rectangle, and we 
have seen how to find the area of any rectangle, but B is 
not quite so simple at first sight. A parallelogram like B, 
which contains no right angle, is called an oblique parallel- 
ogram , 



10 PHYSICS. 

EXERCISE C. 

AREA OF AN OBLIQUE PARALLELOGRAM. 

Apparatus: The 30-ciu. measuring-stick (No. 3). An oblique 
parallelogram of paper about 20 cm. long and 10 cm. wide. (One of 
the straight-edged rulers (No. 24) may prove useful in this Exercise.) 

Draw upon the paper figure a line like c in Fig. 5, taking care 
to make a right angle with the top line 
and the bottom line, and then cut or 
tear the paper along the line c. Take 
the small piece thus removed and join it 
to the larger piece, in such a way as to 




Fig. 5. make a figure that you know how to 

measure. Measure the length and width 
of the figure thus formed and calculate the extent of its surface. 

Then put the two pieces together as they were at first and ask 
yourself whether you could not, if another oblique parallelogram 
were given you, find the extent of its surface without cutting it. 

For the Class-room. 

Estimate without measurement the length and width of some 
visible and convenient rectangles, a book-cover, a table-top, a win- 
dow, etc., and calculate the areas from these estimated dimensions. 
Then take the true dimensions and calculate the true areas. 

Measurement of Volume. 

17. Unit of Volume. — We have now to speak of the 
measurement of volume. The unit of volume may be the 
cubic centimeter, or the cubic inch, or the cubic foot, etc, 
We shall generally use the cubic centimeter as our unit. 

We mean, then, by the volume of a body the number of 
cubic centimeters that could be made of that body if it were 
cut up without waste, as one might cut up a large piece of 
clay or putty. 

18. Rectangular Bodies. — In the case of a body whose 
surface is made up of rectangles, a brick, for instance, it is 



INTBODUCTORY. 11 

easy to see how the volume may be calculated, if we know 
the length and the width and the thickness. We have 
volume = length X width X thickness, 

19. Irregular Bodies.— If the body is of less regular 
shape, like an ordinary stone or a lump of coal, it is not so 
easy to calculate its volume from measurements of length, 
width, and thickness. There is, however, a very easy way 
of finding the volume of such a body by the use of water, 
as will presently be seen. 

20. Volume of Water.— It is easy to find the volume of 
a quantity of water in several ways. One way is to pour the 
water into a rectangular box. Then we can measure its 
length and width and depth and calculate its volume. 
Another way is to pour it into a glass measuring-dish having 
marks upon it to tell the number of cubic centimeters 
required to fill it to certain depths. Another method is to 
weigh the water, for i t is known that one cubic centimeter 
of water weighs one gram. Indeed this is the definition of 
one gram, the to eight of a cubic centimeter of tvater.* If 
the balance which we use for weighing reads in ounces 
instead of grams, we shall have to remember that 1 oz. = 
about 28.3 gm., so that 1 oz. of water will be 28.3 cubic 
centimeters. We shall commonly find the volume of a 
body of water by weighing. 

21. The Water Method. — We will now try the water 
method of finding the volume of a body, a rectangular 
solid. We shall find its volume by the water method and 
also by direct measurement and calculation, and then see 
how well the two results agree. This will test the water 
method, and if we find it to work well, we can use it with 
irregular solids which we cannot measure directly. 

* To be exact one must add at 4° of the centigrade scale of tempera- 
ture. For the purpose of this book such exactness is unnecessary. 



12 



PHYSICS. 



EXERCISE D. 

VOLUME OF A RECTANGULAR BODY BY DISPLACEMENT 
OF WATER, 

Apparatus : A brass can (No. 5) called G in Fig. 6 A small catch- 
bucket (No. 6) called p in Fig. 6. A spring-balance (No. 7). A 
rectangular block of wood (No. 8) so loaded as to sink ia water. 

Closing the overflow tube t of the can G, pour water into G until it 
is filled nearly to the brim. Then open the tube and let all the water 
flow out that will do so, catching it in the small can p, The large 
can should rest steadily upon the table, but the small one is better 
held in the hand when the flow begins, otherwise some water may 
be spilled. The flow should stop rather suddenly at last, with little 
or no drip. 

Throw away all the water thus caught in p and then weigh p on 
the spring-balance to the nearest gram or the nearest twentieth of 
an ounce, according to the graduation of the balance.* Then, 
closing the tube t as before, lower into the can G the wooden block 
until it rests upon the bottom. Then, or sooner if the can G seems 




Fig. 6. 



likely to be overflowed, open the tube t, and as before catch the 
water that runs out in the small can p. The water, Fig. 7, now 

* Ordinary small spring-balances now in the market are often 
marked off in half-ounce divisions, which are about £ inch long. The 
pupil will learn to estimate the position of the pointer when it falls 
between two lines, so as to read to about ^ of au ounce. 



INTRODUCTORY. 



13 



stands just as high in G as it did just before the block was put into 
it. The block has crowded out into the can p just its own bulk of 




Fig. 7. 

water. If, then, we can find the volume of the water that the block 
drove over into p, we shall have the volume of the block itself. 
Weigh p and the water it contains. 

Weight of small can and water = 

11 " " " empty = 

" " water alone = 

If the weight as thus found is in grams, it is equal to the number 
of cubic centimeters in the block. If the weight as thus found is in 
ounces, we must multiply the number of ounces by 28.3 in order to 
find the number of cubic centimeters in the block. 

Now measure carefully the length, width, and thickness of the 
block and calculate the number of cubic centimeters it contains from 
these measurements. 

(Experiments for finding the volumes of irregular bodies by the 
water method may well be postponed till the next Exercise, which 
would otherwise be a very brief one. Potatoes, stones, lumps of 
coal, etc., of suitable size may be used for these further experiments. 



Practice for the Eye. 

A line 10 inches long is drawn on a blackboard with a cross-line at 
any point, and the members of the class estimate the distance from 
either end to the cross-line. Practice like this helps toward accurate 
reading of the spring-balance. 



14 PHYSICS. 



QUESTIONS. 

(1) The true length of a certain line is 16.4 cm. One person meas- 
ures it as 16.6 cm., another as 16.3 cm. How great is the error of 
each in per cents of the true length ? 

(2) A certain rectangle is 50 cm. long and 20 cm. wide. It is meas- 
ured by one person as 50 cm. long and 20.2 cm. wide, and by another 
person as 50.2 cm. long and 20 cm. wide. If the area is calculated 
from each set of measurements, how great (in per cents) will the error 
be in each case ? 

(3) A certain rectangle has a base 100 cm. long and an altitude of 
40 cm. Which will cause the greater error in the estimated area, an 
error of 2 cm. in the base or an error of 1 cm. in the altitude ? 

(4) A rectangular solid is 40 cm. long, 30 cm. wide, and 20 cm. 
thick. How great (in per cents) is the error made by calculating the 
volume from measurements which give 41 cm. for the length, 31 cm. 
for the width, and 19 cm. for the thickness ? 



CHAPTEE II. 

DENSITY AND SPECIFIC GRAVITY. 

22. Definition of Density. — The weight of unit volume 
of a substance is called the density of the substance. If 
we know the density of a substance we can calculate the 
weight of any volume of that substance. Engineers and 
other scientific men often have to find by this method the 
weight of objects which it would be inconvenient to weigh. 
The weights of buildings and bridges, for instance, are 
found in this way. Books used by scientific men contain 
tables giving the densities of many different substances. 

The density of a substance may be expressed as the 
weight in grams of one cubic centimeter, or as the weight 
in pounds of one cubic foot, or in any one of many other 
ways. For brevity, we call the first method of expression 
just given the density in grams and cubic centimeters, and 
the second, the density in pounds and cubic feet. The 
following Exercise will make the matter plainer, and will 
give good practice in measuring and weighing. 

EXERCISE I. 

WEIGHT OF UNIT VOLUME OF A SUBSTANCE. 

Apparatus: A block of wood (No. 9). A spring balance (No. 7). 
A measuring-stick (No. 3). Thread for suspending the block. 

Find the weight of the block in grams and also in ounces. 

Measure the length of each of the four edges which are parallel to 
the grain of the wood, take the average of these measurements and 
call it the length of the block. 

15 



16 P&TS1C3. 

Measure the length of each of the four long edges which are cross- 
wise to the grain of the wood, and call the average of these four 
measurements the icidth of the block. 

Measure the length of each of the four short edges and call the 
average of these four measurements the thickness of the block. 

The weight in ounces is to be turned into pounds. 

From the length, width, and thickne-s in centimeters the length, 
width, and thickness in feet may be found by the rule that 1 ft. = 
30.5 cm., but it is shorter to find the volume in feet from the volume 
in cubic centimeters by the rule that 1 cu. ft. = 28300 cu. cm. 

Calculate, 1st, how many grams, or what part of a gram, 1 cu. cm. 
of the block weighs ; 2d, how many pounds, or what part of a 
pound, 1 cu. ft. of such wood weighs. 

23. Density of Water. — The density of water in grams 
and cubic centimeters is 1; that is, 1 cu. cm. of water 
weighs 1 gm. (see § 20). The density of water in pounds 
and cubic feet is very nearly 62.4; that is, 1 cu. ft. of water 
weighs 62.4 lbs. These numbers for water should be com- 
mitted to memory. 

QUESTIONS. 

1. What ratio is found from the results* of Exercise 1 between the 
density of wood in grams and cubic centimeters and its density in 
pounds and cubic feet? 

2. How does this compare with the ratio of the two densities of 
water, as given above ? 

3. If the ratio is the same for the wood as for water, is this a 
mere coincidence, or is the same thing true in the case of other 
substances ? 

PROBLEMS. 

(1) If a piece of iron 10 cm. long, 8 cm. wide, and 7 cm. thick 
weighs 4000 gm., what is its density in gm. and cu. cm, ? What is 
its density in lbs. and cu. ft. ? 

(2) The density of mercury in gm. and cu. cm. is about 13.6. 
How many lbs. would 1 cu. ft. of it weigh ? 

* It is well to take the average of the results found by the various 
members of the class. 



DENSITY AND SPECIFIC GRAVITY. 17 

24. Weight. — Before going farther we need to think 
carefully about the meaning of the word weight, which we 
have already used a number of times, and shall have to use 
very often. The word has two meanings. 

Sometimes when we speak of the weight of a body we 
mean the amount of the body, as when we speak of 10 lbs. 
of butter or 100 lbs. of iron. 

At other times we mean by the weight of a body the 
amount of the earth's downward pull upon that body, as 
shown by the spring-balance, for instance. 

It is somewhat hard to remember this distinction, 
because the units in which we tell the amount of a body 
have the same name as the units in which we tell the pull 
which the earth exerts upon the body. For instance, we 
say that the earth exerts a pull, ov force, of 5 lbs. upon 
5 lbs. of wood, or 5 lbs. of coal, or anything which consists 
of, or is, 5 lbs. of substance. 

Often when we use the word weight it makes no difference 
which of its two meanings we have in mind, but sometimes 
it does make a difference. Thus, when we put a body 
under water, as we shall do in the next Exercise, and say 
that it appears to lose weight in going from air to water, we 
do not mean that there appears to be any less of the body 
in water than there was in air. We mean that it requires 
a smaller pull of the spring-balance to keep the body from 
sinking in water than it does to keep it from sinking in air. 

25. Mass. — In strict scientific language, the word mass 
is commonly used in speaking of the amount of a substance, 
and the word iveight in speaking of the earth's pull upon 
that substance. For example, a piece of iron the mass of 
which is 50 lbs. is subject to a iveight * of 50 lbs. exerted 
by the earth. 

* Such distinct ions, which use words in a scientific sense different 
from the popular everyday sense, are often necessary in science, but 
it would be rather absurd to try to make the popular use of the 
words agree with the scientific use in all cases. 



18 PHYSICS. 



Specific Gravity. 

26, Definition. — It is often convenient to know the ratio 
which the weight of a body dears to the tueight of an equal 
bulk of ivater. This ratio is called the specific gravity of 
the body. Gravity comes from a Latin word gravis, mean- 
ing heavy. Specific here means distinctive, or particular. 
The specific gravity of a body is its particular heaviness — - 
the degree of heaviness which distinguishes this body from 
other bodies of the same size but different weight. 

27. Loss of Weight in Water. — In finding specific 
gravities it is a common practice to weigh bodies under 
water. The use of this practice will be made plain by 
Exercise 3. The loss of apparent weight suffered by a body 
in going from air to water is shown in Exercise 2. 

EXERCISE 2. 

LIFTING EFFECT OF WATER UPON A BODY ENTIRELY 
IMMERSED IN IT. 

Apparatus: Overflow-can (No. 5). Catcli- bucket (No. 6) Spring- 
balance (No. 7). Loaded block (No. 8). Thread. 

Fill the can and let it overflow and drip as in Exercise D. Catch 
this overflow in the small bucket and throw it away. Then weigh 
the empty bucket in grams. 

Weigh the block in grams before immersing it in the water. 

Lower the block, still suspended from the balance, into the over- 
flow-can till it is entirely covered, catching the overflow and saving it. 

Weigh the block in the water, the balance being entirely above the 
water. 

Weigh the bucket with the overflowed water. 

Subtract the (apparent) weight of the block in water from its 
weight in air, and call the difference the loss of weight of the block in 
water, or the buoyant force exerted upon the block by the water. 

Find weight of the water in the small bucket, and compare this 
with the loss of weight of the block in water. 



DENSITY AND SPECIFIC GRAVITY. 19 

If there is time, make a similar experiment with other "bodies. 

The law illustrated in this Exercise is called from its discoverer 
the law, or principle, of Archimedes. (See any encyclopedia for an 
account of Archimedes.) 

PROBLEMS, 

(1) A certain body weighs 100 gm. out of water and 50 gm. in 
water. How great is the volume of the body? 

(2) A certain body 5 cm. long, 3 cm. wide, and 2 cm. thick weighs 
200 gm. in water. How much does it weigh out of water ? 

EXERCISE 3. 

SPECIFIC GRAVITY OF A SOLID BODY THAT WILL SINK IN 

WATER. 

Apparatus : The spring-balance (No. 7). The gallon jar (No. 10) 
nearJy filled with water. A lump of sulphur (No. 11). Thread. 

Weigh the sulphur out of water ; then in water. 

We know from Exercise 2 that a body immersed in water loses in 
apparent weight an amount equal to the weight of the water whose 
place it has taken. It is easy, therefore, to get from the two weigh- 
ings just made the ratio which we have undertaken to find in this 
Exercise. 

If time permits, find in this Exercise, by the same method that 
is used for the sulphur, the specific gravity of other solids that will 
sink in water — such as glass, coal, etc. 

QUESTIONS. 

1. If the specific gravity of 1 cu. cm. of iron is 7, what is the 
specific gravity of 50 cu. cm. of the same kind of iron ? Of 1 cu. ft. 
of the same kind of iron ? 

2. If the sp. gr. of lead is 11.3, what is the weight in grams of 
1 cu. cm. of lead? What, then, is the density of lead in grams and 
cubic centimeters (see § 22)? 

3. If the sp. gr. of a certain kind of wood is 0.7, what is the weight 
in lbs. of 1 cu. ft. of this wood? What, then, is its density in lbs. 
and cu. ft. ? 

4. A certain body weighs 7 lbs. out of water and 4 lbs. in water. 
What is its specific gravity ? 



20 PHYSICS. 

28, Various Expressions for Specific Gravity. — By defi- 
nition we have 

~ /. 7 7 Wt. of the body 

bp. grav. of a body = 



Wt. of an equal volume of water 

It is evident that the quantity written below the line in 
this definition may be expressed in other ways. We may 
write 

Sp. grav. of a body 

Wt. of the body 
Wt. of water displaced by the body when immersed? 

or 

~ a _ Wt. of the body 

1 ' Loss of weight of the body token immersed 9 

or 

Sp. grav. 

Wt. of the body 
Lifting effect of water upon the body when immersed' 

These expressions all mean the same thing, bat some- 
times one of them is more convenient than the others. In 
the Exercise next before us we shall use the last form. 

EXERCISE 4. 

SPECIFIC GRAVITY OF A BLOCK OF WOOD BY USE OF A SINKER. 

Apparatus]: A rectangular block of wood (No. 9). The spring- 
balance (No. 7). The gallon jar (No. 10) nearly filled with water. 
A lead sinker (No. 12). Thread. 

We have to find two quantities, by experiment : 1st, the weight of 
the body ; 2d, the lifting effect of water upon it ichen immersed. 
Weigh the wood in air and record its weight. 
Now put the block into water, Jou see tliat it floats, fq mtik$ it 



DENSITY AND SPECIFIC GRA VITT. 



21 



stay under water you must hold 
it down. Try this, putting your 
fingers on the block. In this 
case, you see, the lifting effect of 
the water, when the block is 
wholly beneath its surface, is 
greater than the weight of the 
block. We must find out how 
much it is. 

We shalL use the lead sinker to 
hold the block under water, and 
we need to know the weight of 
the sinker alone under water. 
Weigh it in this position and 
record the weight. 

Now suspend the block from 
the balance * and the lead sinker 
from the thread under the block, 
and consider how much the two, 
block and sinker, would weigh in 
the position shown by Fig. 8, the 
block out of water and the sinker 
in water. You can tell this from 
the weighings already made. 
Write it down. 

Wt. of block in air -f Wt. of 

sinker in water =....+.... 

Now lower the 

block and sinker 

till both are cov- 





Fig. 8. 



F*g. 9. 



* The success of a difficult experiment like this 
depends greatly upon the care with which the details 
of the work are thought out by the teacher. The 
following method of attaching the block to the bal- 
ance is recommended : Take a thread two feet long 
and tie the ends together. Then make of it a slip- 
noose by passing one end, I (Fig. 9i, through the 
other end, k. The block may then be placed in the 
noose and the loop I slipped upon the hook of the 
balance, but to prevent slipping when the lead 
weight is to be suspended from the loop below the 
block it is well to pass tlie loop I twice through at z- 



22 PHYSICS. 

ered by the water, and weigh the two together in this position and 
record : 

Wt. of block and sinker together in water = . . . . 

Just before the block entered the water, the sinker being already 
in, the weight was .... Just as soon as the block also was cov- 
ered the weight was only .... The difference is the lifting effect 
of the water upon the block. We have now all that we need for cal- 
culating the specific gravity of the block by means of the formula, 

_ Wt. of block 

y " ~~ Lifting effect of water upon block immersed' 

QUESTIONS. 

(1) A brick-shaped body 20 cm. long, 10 cm. wide, and 5 cm. 
thick weighs 1500 grams. What is its density in gram and centi- 
meter units ? 

What would be the weight of an equal bulk of water ? 
What, then, is the specific gravity of this body? 

(2) A body whose volume is 700 cu. cm. has the density 8 in gram 
and centimeter units. How much does it weigh? What is its 
specific gravity ? 

(3) A body 20 ft. long, 10 ft. wide, and 5 ft. thick weighs 93,600 
lbs. What is its density in pound and foot units ? 

What would be weight of an equal bulk of water, one cu. ft. of 
water weighing 62.4 lbs ? What, then, is the specific gravity of the 
body ? 

(4) A body whose volume is 700 cu. ft. has the density 499.2 in 
pound and foot units. How much does it weigh ? What is its 
specific gravity ? 

(5) What numerical relation do we find in these problems, and in 
those of page 19, between density in gram and centimeter units and 
specific gravity ? 

(6) What relation do we find in the same problems between density, 
in pound and foot units, and specific gravity? 

29. Flotation. — Thus far we have been considering the 
action of water upon bodies entirely immersed in it. We 
shall now have to do with floating bodies. 



DENSITY AND SPECIFIC GRA VII Y. 23 



EXERCISE 5. 

WEIGHT OF WATER DISPLACED BY A FLOATING BODY. 

Apparatus: The same as in Exercise 2, with the exception of the 
sinking body, which is here replaced by one that floats (No. 4). 

Weigh the cylinder, in grams, in air. Find, in grams, the weight 
of water which it displaces from the overflow-can. Compare these 
two weights. 

It will be well to repeat the overflow operation carefully a number 
of times. 

The fact shown in this Exercise concerning the relation 
between the weight of a floating body and the weight of 
water displaced by it should be firmly fixed in the experi- 
menter's mind. It leads to a method of finding the specific 
gravity of floating bodies. 

EXERCISE 6. 

SPECIFIC GRAVITY BY FLOATING METHOD. 

Apparatus : The gallon jar (No. 10) nearly filled with water. A 
slender wooden cylinder (No. 13). A support for holding this cylin- 
der upright in water (No. 14). A measuring- stick (No. 3). 

If a cylinder floated upright with its top just level with the top of 
the water, we should at once know its specific gravity to be 1. If it 
floated just half in and half out of water, we should know its specific 
gravity to be 0.5. The cylinder that we have to use will not float all 
in water or exactly half in water, but if we float it, and find the 
length of tlie part then in the water, we shall, by comparing this 
with the length of the whole cylinder, find some way of ascertaining 
the specific gravity of the cylinder, 

Measure the length of the whole cylinder. 

Float the cylinder in the jar (Fig. 10), keeping it upright by 



24 



PHYSICS. 




means of the holder, which is attached to the side of the jar. Jeggle 

the cylinder to make sure that it 
is free to take its proper posi- 
tion. After each joggling it 
should come to rest at the same 
depth as before. The ruigsof the 
holder must not grip the cylin- 
der at all. When sure that the 
cylinder floats as it should, meas- 
ure the length of the submerged 
part, from the bottom of the 
cylinder up to the flat surface 
of the water. 

To find the specific gravity 
from the two mi asurements 
now made, beg n by recalling 
the fact (see Exercise 5) that 
the water displaced by the float- 
ing cylinder weighs just as 
much as the cylinder itself. 

How many times is the length 
of the submerged part of the 
cylinder contained in the whole 
length ? 

How ma iy times the weight 
of the cylinder would be the 
weight of a like cylinder of 
water ? 

How great, then, do you find 



Fig. 10. 



the specific gravity of the wooden cylinder to be? 

PROBLEMS AND QUESTIONS. 

(1) A block whose specific gravity is 0.6 floats in water. How 
much of it is below the surface ? 

(2) A block whose volume is 1000 cu. cm, and whose specific 
gravity is 0.4, floats in water. How many cu. cm. of the block are 
below the surface ? 

(3) A block that weighs 4 oz. in air is fastened to a sinker that 
weighs 6 oz. in water, and the two together weigh 3 oz. in water. 
What is the specific gravity of the block ? 



DENSITY AND SPECIFIC GRAVITY. 25 

(4) A block whose specific gravity is 0.5, and which weighs 100 
gm. alone in air, is fastened to a sinker that weighs 150 gin. alone in 
water. How much will both together weigh in water? 

(5) A certain body has the density 187.2 in pound and foot units. 
What is its specific gravity ? 

(6) Is the specific gravity of the human body much greater or 
much less than 1 ? 

(7) Why doei filling the lungs w r ith air help one to float in water ? 

EXERCISE 7. 

SPECIFIC GRAVITY OF A LIQUID: TWO METHODS. 

Apparatus: The gallon jar (No. 10) nearly filled with water, and 
the smaller jar (No. 15) nearly filled with a solution of sulphate of 
copper.* The small glass bottle (No. 16). The spring-balance (No. 
7). Thread. 

First Method. 

Weigh the bottle empty. Dip the bottle into the jar of sulphate 
of copper and let it fill with the liquid. Holding the bottle over the 
jar, put the stopper in place, thus crowding out the excess of liquid, 
then wipe the outside of the bottle and weigh it carefully with its 
contents. 

Pour the sulphate of copper back into its jar, then fill the bottle 
with water, just as it was before filled with the other liquid, and 
again w r eigh the bottle and its contents. 

From the three weighings now made the specific gravity of sul- 
phate of copper can easily be found. 

Second Method. 

We found in Exercise 2 that a body going from air into water lost 
in apparent w r eight an amount equal to the weight of its own bulk of 
water. So a body going from air into a solution of sulphate of copper 
will lose in apparent weight an amount equal to the weight of its own 
bulk of the solution. This gives a method of finding the specific 
gravity of the solution. As a body to be weighed first in air, then 
in water, then in the solution, we will use the bottle with enough water 

*This solution may be made by putting 2 lbs. of sulphate of cop- 
per crystals into about 3 qts. of warm water in a glass vessel and 
stirring occasionally till the crystals are dissolved, 



26 PHYSICS. 

in it to make it sink in either liquid. We may, indeed, use the bottle 
full of water, just as it was left at the end of the first part of this 
Exercise. 

EXPERIMENTS. 

1. Exhibit and show in operation two graduated glass hydrometers 
— one for determining the specific gravity of liquids less dense than 
water (App. No. XI), the other for use with liquids more dense than 
water (App. No. XII. \ 

2. Show in a bottle together several liquids of different specific 
gravities that do not tend to mix with each other ; for instance, mer- 
cury, chloroform, water, and kerosene. 

3. Take a small tumbler containing some mercury and drop into 
it a piece of iron. Do not put into it gold or silver, as mercury at- 
tacks these metals. 

4. Place a dry sponge on water. It floats lightly, but is the spe- 
cific gravity of the fibres of the sponge greater or less than that of 
water ? To answer this question push the sponge beneath the sur- 
face. What rises from it ? Squeeze the sponge very hard till noth- 
ing more seems to come from it. Now will it rise to the surface 
when released ? 

QTJESTIONS. 

1. A glass sphere which weighs 100 gm. in air weighs 60 gm. in 
water and 40 gm. in sulphuric acid of a certain strength. What is 
the specific gravity of the glass ? 

What is the specific gravity of the sulphuric acid? 

2. A vessel contains a layer of water 10 cm. deep and above this a 
layer of kerosene (sp. gr. 0.8) 10 cm. deep. What is the weight of 
a cube, each edge of which is 10 cm. long, that, if placed in this ves- 
sel, will sink till one-half its volume is in the water and one-half in 
the kerosene? Ans. 900 gm. What is its specific gravity? Arts. 
0.9. 

3. A certain ship weighs with its cargo 10,000 tons. 

(a) How many cubic feet of fresh water would it displace ? 

(b) How many cubic feet of sea- water of specific gravity 1.026 
would it displace ? 

4. If 1 cu. cm. of mercury weighs 13.6 gm. and 1 cu. cm. of cork 
weighs 0.25 gm., how deep will a cylinder of cork 20 cm. long sink, 
when placed on end in mercury ? 



DENSITY AND SPECIFIC GBAVITT. 27 

5. Which has the greater specific gravity, cream or skimmed 
milk? 

6. A piece of cloth thrown upon water will float at first and after- 
ward sink. Why does it not sink at once ? 

7. Can the pupil tell from his own observation which of the follow- 
ing substances are more dense and which are less dense than water : 
kerosene-oil, ordinary lubricating- oil, butter, cheese, potatoes, eggs, 
meat, ice, india-rubber ? 



CHAPTEE III. 

FLUID-PRESSURE. 

30. Fluids. — Water and air readily floiv from one position 
or shape to another. They are examples of that class of 
substances called fluids. Fine sand and other like sub- 
stances flow in a certain way, but examination shows them 
to consist of little hard or tough particles very different 
from equally small particles of water. 

Fluids are divided into liquids and gases. Water is an 
example of the liquids ; air an example of the gases. 

31. Fluid-pressure. — Fluids settle snugly around solid, 
that is, non-fluid, bodies placed in them and act upon these 
bodies with a peculiarly even pressure. We shall now make 
some experiments with liquid-pressure and later with gas- 
pressure. 

EXPERIMENTS WITH PRESSURE-GAUGE. 

Fill the gallon glass jar (No. 10) with water to a level about one 
inch from the top. Close the smaller end of a student- lamp chim- 
ney tight with a good cork stopper. Make the pressure-gauge (No. 
I.) ready for use by the following operation, having first put on a 
fresh rubber diaphragm if necessary : Release the glass tube from 
the rubber tube and wet the whole length of the glass tube inside 
with water, leaving within it a column of water about one-half inch 
long to serve as an index.* Hold the gauge itself under water for a 
little time before reconnecting the glass tube with the rubber tube, 

*It may be necessary to use water colored by some aniline dye be- 
fore a large class. 

28 



FLUID-PRESSURE. 29 

in order to allow the air within the gauge to come to the temperature 
of the water. On reconnecting the glass tube leave the water-index 
near the rubber tube. 

Different Letels. — Xow push the gauge down into the jar and 
raise and lower it repeatedly in the water, keeping the glass tube 
with the water-index horizontal, and let the class determine from the 
movements of this index whether the pressure of the water against 
the rubber diaphragm increases or decreases when the gauge is 
pushed deeper in the water. 

Different Directions. — Rest the bottom of the supporting pillar 
of the gauge upon the bottom of the jar, and, still keeping the glass 
tube horizontal, turn the upper pulley so that by means of the rubber 
band the lower pulley will be turned and the rubber diaphragm will 
face downward, sidewise, and upward in succession, its centre re- 
maining practically unchanged in position. Let the class determine 
by watching the water-index whether the pressure upon the rubber 
diaphragm is any greater when it faces upward than when it face? 
downward or sidewise. 

Different Points on the S^me Leyel. — Push the closed end 
of the lamp-chimney down into the water till it is near the bottom of 
the jar. Move the gauge face about, without changing its level, so 
as to bring it under this closed end. Move it now out of and now into 
this position, thus changing the depth of water immediately above it 
from one-half inch or less to several inches. Let the class determine 
by watching the index whether such changes of position, without 
change of level, make any difference in the pressure against the gauge- 
face. 

We shall make considerable use farther on of the facts 
brought out by these experiments. Just here we can see 
that they explain, at least in a general way, why a body 
immersed in water weighs, or appears to weigh, less than 
when in the air. For we see that there is an upward 
pressure of the water against the under side of the body, 
and that this upward pressure is greater than the down- 
ward pressure against the upper side of the body. 

32. Slight Effect of Pressure upon the Density of Water. 

— Having seen that there is greater pressure on low levels 



30 PHYSICS. 

than on high levels in water, we may well ask whether this 
greater pressure crowds the particles of water closer together 
on the low levels, thus making the water denser than on 
high levels. In fact there is an effect of this kind, but it 
is so slight that we need take no account of it in any ordi- 
nary case. It is very difficult to compress water much. 

EXPERIMENT. 

Fill a bottle with water and close it with, a rubber stopper having 
one hole through it. Then, holding the stopper firmly in place, push 
down into the hole a solid brass rod of a size to fit rather closely. 
The bottle will probably be broken by this effort to compress the 
water within it. (App. No. II.) 

33. Uniform Increase of Pressure with Depth. — We have 

not made, and cannot well make with the gauge used, any 
accurate measurement of the rate at which pressure changes 
with change of level in water. The fact is, however, that 
if we place a surface of 1 sq. cm. horizontal at any depth 
in water the column of water just above it is resting upon 
the given surface.* If we carry the given surface down 
1 cm. farther, we now have resting upon it a load somewhat 
greater than before, greater by the weight of the additional 
1 cu. cm. of water which is now above it. As 1 cu. cm. of 
water weighs 1 gm., the pressure upon a surface of 1 sq. 
cm. changes by 1 gm. for each 1 cm. change of level in the 
water. 

QUESTIONS. 

A cubical box, 10 cm. along each edge, has extending from its top, 
as in Fig. 11, a tube 15 cm. tall and 1 sq. cm. in cross-section (inside). 

(1) If the box, but not the tube, is full of water, how great is the 
water-pressure on the whole of the bottom ? 

* The pressure upon the given surface may be greater than the 
weight of the column of water resting upon it, for there may be, and 
usually is, a downward pressure of air or something else upon the 
top of the water-column. 



FLUIb-PRBSStmE. 



31 



(2) If the tube as well as the box is full of water, how great is the 
pressure upon that one sq. cm. of the bot- 
tom which lies just beneath the tube ? 

(3j Is the pressure equally great per sq. 
cm. at other parts of the bottom ? 

(4) How much is the total pressure now on 
the bottom of the box? 

(5) How great is the pressure per sq. cm. 
at the top of the box just at the bottom of 
the tube ? 

(6) How great is the total upward pressure 
of the water against the top of the box ? 

(Disregard the atmospheric pressure upon 
the top of the water-column in all these 
questions at first. Afterward call this at- 
mospheric pressure 1000 gm. per sq. cm., 
and ask the same questions as before.) 

34. Gas-pressure. — We have made 
some experiments with liquid-pressure. 
We must now begin to learn some- 
thing about air-pressure, which in many practical matters 
of e very-day life has a very important connection with 
water-pressure. We will at the start repeat in a slightly 
varied form a famous experiment first made by Torricelli, 
an Italian, about the middle of the seventeenth century. 
It is intended to show the pressure of the air about us, 
which is called atmospheric pressure. 



Fig. 11. 



EXPERIMENT.* 

Take two pieces of strong glass tubing about 0.7 cm. in inside di- 
ameter, one of them, about 1 m. long, closed at one end, and the other, 
about 20 cm. long, open at both ends, and connect them by means of 
a thick- walled piece of rubber tubing about 25 cm. long. The rubber 
tube should fit tight upon the glass tubes, and for greater security 
should be fastened on by means of wire or string. 

* This experiment can be more conveniently performed with a 
single straight glass tube if a mercury-well is available. 



32 PHYSIOS. 



Holding the tubes thus connected (A pp. No. III.) by the 
free end of the short glass tube, the closed end of the long 
glass tube hanging down, pour mercury by means of a small 
funnel of glass or paper into the tubes, tapping or shaking 
them occasionally to dislodge air-bubbles, until the top of the 
mercury-column reaches the rubber tube. Then gently raise 
the closed end of the long glass tube until this tube points 
straight upward (Fig. 12), meanwhile holding the other 
glass tube upright and taking care that no mercury is spilled. 

During the latter part of this operation it will be noticed 
that the mercury begins to fall away from the closed end 
of the long glass tube, and finally several inches of this 
tube will be apparently empty.* But the mercury contin- 
ues to stand very much higher in the long glass tube than 
in the short one. 



y 



Fig. 12. 

35. Explanation. — It was known before the time of 
Torricelli that if air was drawn from the upper part of a 
tube the lower end of which rested in water the water 
would rise in the tube, but the true reason for this was not 
known. Torricelli maintained, and Pascal, a Frenchman, 
showed by experimenting at different heights in the air, 
that the pressure of the atmosphere, due to its weight, 
accounted for the rise of liquids in a vacuum. We have 
only to think of the fact that the air, although its density 
is very small compared with that of water, has, because of 
its great quantity, a great weight, and we see that the air, 
pressing upon the mercury-surface in the shorter tube, 
balances the column of mercury in the long tube. 

36. Amount of the Atmospheric Pressure — Barometer. — 

By measuring the difference in height of the two mercury- 

* Really this space contains a very little air, from the bubbles that 
were in the mercury-column before it was inverted, but so little that 
we may at present disregard it and consider the space above the mer- 
cury as empty. Such a space is called a vacuum, from a Latin word 
meaning empty. 



FL TJID-PBESS USE 



u 



surfaces we can get a measure of the atmospheric pressure. 
We find that the atmospheric pressure is about as great 
upon the surface of the earth as would be the pressure of a 
layer of mercury 76 cm. deep, or a layer of water about 
10.3 m. deep, over the whole earth. The pressure per 
square centimeter at any given part of the earth's surface 
varies somewhat from day to day, and even from hour to 
hour. 

If we fasten the apparatus that has just been used to a 
suitable support, it will serve permanently as a rude 
barometer, indicating the variations of the atmospheric 
pressure. 

37. Pressure in Different Directions. — Air-pressure, like 
liquid-pressure, is at any given point equal in all direc- 
tions, if the air is at rest. 



EXPERIMENT. 



J- 



-J 




Take a strong thistle-tube (No. IV.) of 
the shape shown in Fig. 13 and tie a piece 
of thick sheet rubber across the mouth, 
which may be about 1 inch in diameter. 
Make the covering air-tight by means of 
some cement, melted beeswax and rosin, 
for instance, poured in at the point J. Con- 
nect this thistle-tube by means of a thick- 
walled rubber tube to an air-pump (No. 
V.), and exhaust the air. The rubber cap, 
not being supported by air-pressure beneath, 
will now be pushed down by the atmos- 
pheric pressure into a deep cup-shape 
Pinch the rubber tube so that no air shall 
leak back into the thistle-tube, and then 
turn the mouth of the latter in all directions, 
sidewise, downward, and oblique. Observe 
whether the depth of the rubber cup 
changes during this operation, as it would do if the pressure upon 
it changed. 



Fig. 13. 



34 PHYSIOS. 

38. Air-pressure at Different Levels. — We should find 
by proper experiments that in air at rest, as in water at 
rest, pressure is equally great at all points on the same 
level. We should find, also, that the air-pressure diminishes 
with increase of height from the earth's surface, but, as 
the density of air is very little compared with that of 
water, it requires a considerable change of level to make 
much difference in the air-pressure. 

The rate at which atmospheric pressure decreases with 
increase of height being well known, it is a common prac- 
tice to estimate the height of mountains by noting the 
difference of atmospheric pressure at the summit and base. 
"Aneroid " barometers are frequently used for such work. 
Aneroid means without liquid. An aneroid barometer 
contains no mercury nor other liquid. It is an air-tight 
metal box with a flexible metal cover. The middle of the 
cover moves in or out slightly with changes of pressure, and 
its slight motions are magnified to the eye by various 
mechanical contrivances. Some aneroid barometers are 
about as large as ordinary watches and look much like 
them. 

39. Difference between Liquids and Gases. — Liquids are 
much heavier than gases, in most cases. Most liquids are 
easily seen. Most gases are practically invisible. But per- 
haps the most striking difference between liquids and gases 
is a difference in compressibility. We have seen that it is 
difficult to compress water much, but it is very easy to com- 
press air. 

EXPERIMENT. 

Take the bent glass tube (No. VI.), closed at one end, and pour 
into it a little mercury, enough to fill the bend. At first the mercury 
will stand a little higher in the long arm, but by tipping the tube 
and letting ont a little of the air imprisoned in the short arm the level 
can be made nearly the same in both arms, as in Fig. 14. Now 



FLUID-PRESSURE. 



35 



measure the length of the imprisoned air-column, and write it under 
the letter V* on the blackboard. 



P. 



FXP. 



Fig. 14. 



The pressure upon this air is now, if the mercury- level 
is the same in both arms, equal to that upon the unim pris- 
oned air. It is as great a pressure as would be exerted by 
the weight of a column of mercury as tall as that in the 
barometer (Fig. 12). Take, then, a reading of this barom- 
eter and record this reading under the letter P. 

Pour in more mercury till the difference of level in the 
two arms is about 20 cm., then measure again the length 
of the inclosed air-column. Record this length under V, 
and record under P the present difference of mercury 
level plus the height of the barometer column. 

Proceed by stages in this way till the volume of the inclosed air- 
column is about one-half what it was at first. Multiply each number 
under Fby the corresponding number under P, and write the prod- 
ucts in the column headed VX P. 

40. Boyle's Law. — An examination of the last column 
in the table of the preceding section will probably indicate 
a very simple law connecting pressure and volume in the 
case of a given body of air. This law is important, and 
should be remembered by the pupil. It is sometimes 
called Boyle's law and sometimes Mariotte's law. "We shall 
call it by the shorter name, Boyle* s lata. 

Illustrations and Applications of Fluid-pressure. 

41. Principle of the Hydraulic Press. — The questions on 
pp. 30 and 31 have brought out the fact that pressure trans- 



* The length of the air-column is the same as its volume, if we take 
for our unit of volume the space contained in unit length of the tube. 



36 physics. 

mitted through a small tube may extend to a broad surface 
beyond the tube so as to make the total pressure on this 
surface very great. The following experiment will show 
that similar effects can be produced with air-pressure. 

EXPERIMENT.* 

Take a common rubber football and blow air into it till it is about 
half filled, connecting a rubber tube with the key for greater con- 
venience in blowing (App. No. VII). Then rest one end of a board, 




Fig, 15. 

f S in Fig. 1 5, on the football and the other end upon a box or block 
of about the same he : ght. Then place a weight of 25 lbs. or more 
on the board nearly over the ball, holding the rubber tube attached 
to the key in such a way that the air cannot escape from the ball. 
Then blow through the tube into the ball and observe that you can 
in this way lift the weight. 

42. Hydrostatic Press. — The preceding experiment 
illustrates the operation of the hydrostatic press, a machine 
in which a very great force is obtained, for lifting or 
compressing bodies, by pumping water through a small 
tube into a large cylinder, one end of which is closed by a 
movable stopper called & piston. (See § 204.) 

EXPERIMENTS. 

1. Take again the pressure-gauge and the accompanying apparatus, 
§ 31. Fill the lamp-chimney with water, and then, holding a card 
across the open end, invert the chimney, lower the end covered 
by the card into the water, and then remove the card. Most of the 
water will now remain in the chimney, although its upper end is nine 
or ten inches above the surface of the water in the jar. 

* An experiment with Gage's piston and cylinder apparatus may 
be substituted for this to show the same effect. 



FL UID-PRESSURE. 



37 



How does tlie pressure per sq. cm. inside the chimney on a level 
with the outside water surface compare with the pressure per sq. 
cm. at this outer surface, that is, the atmospheric pressure ? 

How, then, will the pressure per sq. cm. at points higher in the 
chimney compare with the atmospheric pressure ? 

After these questions have been answered by the aid of what the 
class already knows about liquid-pressure, test the correctness of the 
answer by means of the gauge. 

2. Take a long narrow glass tube open at both ends, and dip one 
end into a vessel of water. Apply the lips to the other end and 
draw the water up till the tube is filled. 

In what sense is the water drawn up ? 

(The operation begins with an expansion of the lungs which les- 
sens the air-pressure within them. Then air runs from the place of 
high pressure, the tube, to the place of low pressure, the lungs. So 
the air-pressure within the tube is lessened.) 

3. After nearly filling the tube as in Experiment 2 quickly close 
the top with a finger and then lift the lower end from the water. 
Uncover the top of the tube for an instant, then cover it again. 

Explain the behavior of the water during these operations. 

4. Fill or nearly fill a tumbler or broad-mouthed bottle with water 
and then cover it with a sheet of thick paper. Hold the paper firmly 
in place with the hand and invert the tumbler ; then take away the 
hand that holds the paper. (As accidents may happen, the tumbler 
should be held over some large dish.) 

In this experiment it should be noticed that 
the paper does not press close against the rim of 
the tumbler after the inversion. It hangs rather 
loose, having dropped down or sagged a little, 
thus allowing the air above the water to expand 
a trifle, decreasing in pressure. 

5. Fig. 16 (App. No. VIlIj shows a bottle 
closed with a rubber stopper through which 
two glass tubes, a and b, open at both ends, ex- 
tend. To one of the tubes, a, is attached a rub- 
ber tube, r. The bottle and the two glass tubes 
are full of water. 

By applying the lips to the outer end of the 
tube r water can b • " drawn " into the mouth, 
when the tube b is closed by a finger at the top? 



a 



\ 



Fig. 16. 
('an this be done 



38 



PHYSICS. 



6. Show some form of the Cartesian diver (No. XIV), explaining 
why it sinks when greater pressure is put upon the water in 
which it is placed. 

43. Pumps. — Many contrivances for making fluids run 
from one place to another are called pumps. A flow may 
be caused by decreasing the pressure at the place where the 
fluid is to be delivered or by increasing the pressure at the 
place from which it is to be removed. 



EXPERIMENT. 

Show in operation glass models of the "lifting-pump" (App. 
No. IX, Fig. 17) and "force-pump" (App. No. X, Fig. 18), dis- 
cussing their action. 



L 



u 



flflfi 



¥ 



© 



[ 



D 



\ 



or 



Fig. 17 



Fig. 18. 



FLUID PRESSURE. 



39 



^4 



44. The Siphon. — The apparatus illustrated in the fol- 
lowing experiment is called the siphon. It is found in a 
great variety of forms and is of much use. 

EXPERIMENT. 

Take two glass tubes, each about 6 in. long, connected by a rubber 
tube about 1 ft. long. Fill the whole with water, then close each 
end with a finger. Hold one end be- 
neath the surface of the water in the 
gallon jar (Fig. 19) ; remove the fin- 
ger from that end, and bring the 
other end, still closed, down outside 
the jar to a level lower than the water 
surface. 

Is the water-pressure against the 
finger that closes the tube now greater 
or less than the atmospheric pressure 
upon an equally large surface? If 
greater, the water will run out when 
the finger is removed. If less, the air 
will run in and drive the water up in 
the tube when the finger is removed. 
Try the experiment. 

Repeat the experiment, but now 
hold the outer end of the tube, before 
opening it, higher than the level of the water in the jar. 

45. Balancing Columns. — The method of finding specific 
gravities that is suggested by the following experiment is 
called the method of balancing columns. In the form here 
shoAvn it cannot well be used with liquids that naturally 
mix with each other, as alcohol and water do. Later this 
general method will be used in a form that does not bring 
the two liquids into contact with each other. 



Fig. 10. 



EXPERIMENT. 



Take a bent glass tube (App. No. XIII, Fig. 20) each arm of which 
is about one foot long, and pour water into it till both arms are 



40 



PHYSICS. 



Fig. 20. 



about half full ; then pour kerosene into one arm till it is nearly full. 
Does the water now stand as high in the other arm as 
the kerosene does in the first arm ? Can you from this 
experiment see a third method for finding the specific 
gravity of a liquid ? 

QUESTIONS. 

(1) Does water stand at the same level in the spout as 
in the main part of a watering-pot? 

(2) If one branch of a U tube (see Fig. 20) were larger 
than the other, would water stand at the same level in 
both? 

(3) Is it necessary in finding the specific gravity of a 
liquid by the method indicated in Art. 45 to have the 

two branches of the tube equally large ? 

(4) Does the height of mercury in the tube of a barometer depend 
upon the size of the tube? (We neglect at present w T hat is called 
the capillary effect. See Second Part.) 

(5) If the height of the barometer mercury-column, of specific 
gravity 13.6, is 76 cm., how tall a column of water could be sustained 
by the atmospheric pressure if there were a vacuum above the 
water ? (Give the answer in ft. as well as in cm.) 

(6) A water-tank 10 ft. deep is to be emptied by means of a tube 
used as a siphon. What is the least length the tube can have? 

(7) With ordinary atmospheric pressure what is the greatest height 
to which water may be raised by means of a pump working above it? 

(8) Do you understand the operation of the " trap" which allows 
water to flow from a sink to a sewer, but does not allow gas to come 
from the sewer to the sink ? 



CHAPTER IV. 

THE LEVER. 

46. Definition and Illustration. — Civilized men do most 
of their work with tools or machines. Many tools and 
many parts of machines consist of a piece of iron or wood 
or other material movable to a certain extent upon a sup- 
port called a pivot, or axis, or fulcrum, by means of which 
a force applied in one direction at a certain spot may pro- 
dace another force different in direction or in magnitude, 
or in both, at another spot. Such a tool or part of a 
machine is called a lever. 

One of the most familiar examples of the lever is a crow- 
bar. A hammer, as used to draw out a nail from a board, 
is another example. Each half of a pair of scissors is a 
lever. We shall study some very simple forms of the lever 
to find out what relations hold between the forces exerted 
at different points. 

EXERCISE 8. 

THE STRAIGHT LEVER: FIRST CLASS. 

Apparatus : The lever and supporting bar (Xo. 17) fastened to the; 
long horizontal bar that reaches above the table from end to end. 
Two scale-pans (Xos. 18a and 18b). A set of weights (No. 19). 

Hang one scale-pan carrying a load of 8 oz. on the right-hand end 
of the lever at a distance of 14 cm. from the middle, as in Fig. 21. 

Hang the other pan, with an equal load, on the left-hand end of 
the lever, at such a distance from the middle that the lever will bal- 
ance, that is, stay horizontal when once placed so, even when the ap- 

41 



42 



PHYSICS. 



paratus is jarred somewhat by tapping the short bar to which the 
lever is attached. Then make a record like this : 



Right dist. 
fr. centre. 



Left wt. fr! centre Ri ^ ht wt - 

(l+8) = 9oz. (1 + 8)= 9 oz. 14.0cm. 

(The space here left blank (in the record) is to be filled by the left- 
hand distance which the student finds necessary to make the apparatus 
balance.) 

Change the right-hand weight to 7 oz. , keeping its place unchanged, 
and move the left-hand weight, still 9 oz., to some new position 
which will make the whole balance, in spite of jarring as before. 
Make a record, as before, of the weights and distances, putting it just 
beneath the record for the first arrangement. 

Change the right-hand weight to 5 oz. without changing its place, 
and find what position the left-hand weight, still remaining 9 oz., 
must have in order that the lever may balance. Record the distances 




Fig. 21. 

and weights for this case under the records already made for the 
first and second cases. 

One more case may be taken, in which the right-hand weight be- 
comes 4 oz., still at 14 cm., which will give a fourth line in the record 
table. More observations with different arrangements might be 
made, but it is better to make a moderate number of good observa- 
tions than a large number of hasty or careless ones, 



THE LEVER. 



43 



By studying the record table now made the student should find a rule 
by which, when the two weights and one distance are given, the other 
distance can be found by calculation ; or when the two distances and 
one weight are given, the other weight can be found by calculation. 

QUESTIONS. 

(1) If a mass of 6 oz. is suspended from a point 4 cm. to the left of 
the centre of the lever in Exercise 8, how great a load placed at a 
distance of 10 cm. to the right of the centre will make equilibrium ? 

(2) A mass of 8 oz. is suspended from a point 5 cm. from the centre 
of the lever and is balanced by a mass of 10 oz. How far from the 
centre is the latter placed? 

(3j Two masses, 4 oz. and 12 oz. respectively, are to be suspended 
from a lever. Describe three possible arrangements of the masses, 
any one of which will cause them to balance. 

(4) A boy pushing down at one end of a lever 6 ft. long pries up a 
stone weighing 100 lbs. at the other end. The fulcrum is 2 ft. from 
the stone. The weight of the lever itself is neglected. How great is 
the force exerted by the boy ? 

47. More than Two Weights. — In the preceding Exer- 
cise the class found out how to 
make the two weights hung [ 
from the lever balance each 
other. Let us ask now what the 
rule for balancing would be if 
there were more than two 
weights in use, as in Fig. 22, for 
instance. 

EXPERIMENTS. 

We will make the apparatus balance with four weights, two on 
each side. We will call the weight nearest the centre on the left 
hand weight Xo. 1, which we will write TF,, for short. The other 
weight on the left-hand side we will call Xo. 3, or TT 3 . 
weights on the right hand we will call Wz and TF 4 . 
When the whole balances, we will call 

The distance of Wi from the middle, D x , 
" " W% " " " D* t 

" u M\ " " " D<. 



tt3 tt 



Fig. 22. 



m 



The two 



44 



PHYSICS. 



Now if we go back for a moment to the case of two weights, which 
the class has studied, and if we call these Pi and P 2 , and their dis- 
tances from the middle d x and d 2 , we can state the rule for balancing 
in this way : 

Pi X ^1 must equal P 2 X d 2 . 

In the new case, where we have four weights, we may guess * that 
the rale is 

(Wi X A) + (TT 8 X P 3 ) = (TF 2 x D 2 ) + (TF 4 X B A ) 9 

and then test the truth of our guess by trial. 

Try other like cases. 

48. Circular Lever. — In the experiments with which we 
have just been engaged the weights have been suspended 
from the top of the lever on a level with that part of the 

pivot upon which the lever rests. 
In other experiments which are 
to follow we shall not always be 
able to keep this arrangement, 
I and we have now to find out what 
' would be the effect of hanging 
one or more of the weights from 
points higher or lower than the 
point of support of the lever. 
For this purpose we shall use No. 
FlG - ~ 3 - XV, the piece of apparatus 

shown in Fig. 23, in which the straight lever thus far used 
is replaced by a circle of wood about 8 inches in diameter, 
supported by a screw passing horizontally through the 
centre. Such a circle, or disk, of wood comes under the 
general definition of a lever. 

EXPERIMENTS. 

We will hang at b and / such weights as will balance each 

* Shrewd guessing, followed by a test, should be encouraged by the 
teacher as a means of extending knowledge. In fact, it is the con- 
stant resource of the investigator. 




THE LEVER. 



U 



other, leaving the disk in equilibrium, and will then move one of 
the weights to a point vertically above or vertically below its pres- 
ent place ; that is, from /to e or g, or from b to a or c. Shall we still 
have equilibrium ? Try. 

We will now turn the disk a little, so that the lines ab c and efg 
will be no longer quite vertical, and will see whether now a weight 
at e or at g has just the same effect as if at/. Try. 

A careful note should be made of conclusions for future use. 

49. Centre of Gravity. — In the experiments upon the 
lever thus far, the lever itself, whether a bar or a disk, has 
balanced, when left to itself without load. We have, 
therefore, not had to consider the weight of the lever itself. 
But many levers are used in such a way that their own 
weight helps or hinders the operation to be performed with 
them. To understand such cases we must learn something 
about what is called the centre of gravity of a body. 

EXPERIMENT. 
Take a board, cut in any irregular shape, like Fig. 24, for 




instance. 



Fig. 24. 
"Bore several small holes straight through the board, 




46 PHYSICS. 

and put into each hole a wire nail that will fit close, long enough 

to project about half an inch on each side of the b ard. Tie a 
bullet at one end of a thread and make a loop in the other 
end. Put this loop over one hook of a piece of wire bent 
into the shape shown in Fig. 25, and then rest the nail a, 
Fig. 24, in the hooks of the same wire, so that the board and 
the string carrying the bullet will both hang free, the string 

Fig. 25. near the face of the board.* Mark with a pencil the course 

of the string downward across the board. 

Then suspend the board by the nail b and mark the new course of 

the string. Proceed in this way with all the nails and note the point 

where the various pencil-marks cross each other. 

Finally, place the board horizontal and balance it upon the flat head 

of a lead-pencil, noting how near the head of the pencil comes to the 

crossing of the lines marked on the board. 

Definition. — By such experiments as this we come to see 
that there is within the board a certain point which always 
hangs just beneath the support when the board comes to 
rest suspended from any one of the nails. We see that the 
same point has to be just above the support when the board 
rests upon the pencil-top. In short, the board acts in 
these experiments as it would if all its weight were concen- 
trated at this particular point. This point might be called 
the centre of weight or centre of heaviness of the board, 
but it is commonly called the centre of gravity. 

50. Weight of the Lever. — The following Exercise is in- 
tended to make the pupil more familiar with the idea of 
centre of gravity, and to show how it may be taken account 
of in the use of the leyer. 

EXERCISE 9. 

CENTRE OF GRAVITY AND WEIGHT OF A LEVER. 

Apparatus : The lever of No. 17, detached from its supporting 
bar, and a small block (No 21), the two being fastened together, as in 

* The whole apparatus as shown in Fig. 24 will be called No. XVI.) 



THE LEVEB. 47 

Fig. 26, so as to make one body, the whole of which will be called 



Fig. 26. 

the lever in this Exercise. A slender wooden cylinder (No. 13). A 
1-oz. scale-pan (18 A or 18 b). A 1-oz. wt. from No. 19. 

To find the centre of gravity of the lever, balance it as nearly as 
you can, bar and block fastened together, in a horizontal position on 
the cylinder laid on the table (see Fig. 26), the cylinder being kept 
at right angles with the lever. Find in this way at what particular 
mark of the bar the centre of gravity is, and record this mark — for 
example, 9.1 cm. 

Then suspend the 1-oz. scale-pan carrying a 1-oz. wt. , 2 oz. in all, 
from any convenient point near the free end of the bar, and letting 
this end project beyond the edge of the table-top, balance the whole, 
as now arranged, as nearly as you can, on the cylinder laid on the 
table as before (see Fig. 27). 

Now record the mark from which 
the scale-pan hangs, 33.4 cm., we 
may suppose, and the mark which 
is just over the middle of the cyl- 
inder when the whole balances, 
21.6 cm., let us say. 



ZDZ 



This case is like that of the lever 
studied in Exercise 8. The cylin- p IG# 27. 

der now taking the place of the screw as a support, we see that 
the left-hand weight is 2 oz., 
" " " distance is 33.4 -21.6 = 11.8 cm., 
" right-hand weight is the weight of the lever, 
" " " distance is 21.6 -9.1 = 12.5 cm., 

that is, the distance from the support in Fig. 27 to the centre of grav- 
ity of the bar and block. 

We do not as yet know the weight of the lever, but we will call it 
Wi> and see whether we can find its amount by calculation. If we 
apply the same rule that was found to hold true in Exercise 8, we 
shall have 

2 X 11.8 = W, X 12.5, 



48 PHYSICS. 

which gives for the weight of the bar and block 

2 X 11.8 
Wi = ,. - ' =1.89 oz., nearly. 

The value of TT 2 obtained in this way by the pupil should be com- 
pared with the weight of the bar and block as found by the teacher 
with some balance, e.g. No. XVII, much more sensitive than the 
spring-balance used by the class; for if the method of this Exercise is 
carefully followed it will give the weight of the lever more accurately 
than the spring-balance is likely to do. 

QUESTIONS. 

(1) An oar, the centre of gravity of which is 3 ft. from the end of 
the handle, weighs 4 lbs. It rests in the rowlock at a point 2 ft. 
from the end of the handle. How great a force applied at the end of 
the handle will keep it balanced? 

(2) A boy weighing 100 lbs. is see-sawing alone on a plank 20 ft. 
long weighing 50 lbs. The boy's centre of gravity is 1 ft. from one 
end of the plank. How far from the same end of the plank is the 
fulcrum ? (The centre of gravity of the plank is at its middle). 

(3) If the plank mentioned in the preceding problem is to balance 
on a fulcrum 8 ft. from one end, with a 100 lb. boy 1 ft. from this 
end and another boy 1 ft. from the other end, how much must the 
second boy weigh ? (See Art. 47). Ans. 54 T 6 T lbs. 

51. Remarks. — We have now found out how to take 
account of the weight of the lever itself, when we need to 
do so. We know that all its weight may be regarded as 
concentrated at a certain point, which we call the centre of 
gravity, and we have tried one case in which the weight of 
the lever itself, acting at the centre of gravity, balanced a 
certain weight suspended from the bar. In common levers, 
like the crowbar, the weight of the bar itself is sometimes 
very important, when the f alcrum is a long distance from 
the centre of gravity of the bar. 

Centre of gravity will be taken up again, and the differ- 
ent kinds of eqtiilibrium, stable, unstable, and neutral, will 
be considered in the Second Part. 

We will now return for the present to cases of the lever 



THE LEVER. 



49 



where the centre of gravity lies, as in Exercise 8, just under 
the point of support of the bar. In such cases the weight 
of the lever itself does not tend to make the bar tip in 
either direction from its horizontal position. 

Classes of Levers. 

52. Lever of the First Class. — In the levers which we 
have studied thus far the support, or fulcrum as it is often 
called, lies between the lines of suspension of two weights. 
This kind of lever, whether it is a simple bar or a disk or 
an object of irregular shape, whether its centre of gravity 
lies at the point of support or not, is called a lever of the 
First Class. 

53. The Power, Power-arm, etc.- — To take a simple and 
convenient case, we will consider in Fig. 28 a circle sup- 
ported at its centre, F. We 
will suppose that this lever is 
used for the purpose of support- 
ing a weight TF, and the force 
used for this purpose, whether 
it is applied by means of another ! 
weight, as in the figure, or by 
means of the hand, or in any 
other way, we will call the 
power. 

We have seen in the experi- 
ments of § 48 that, as the 
lever now stands, it makes no 
difference whether W is sus- 
pended from the point which now carries it or from some 
point higher or lower in the same vertical line, which is 
called the line of action of W. A like statement can be 
made for P. We shall call the shortest distance from P 's 
line of action to the fulcrum the power-arm, and the short- 




Fig. 28. 



50 



physics. 



est distance from JF's line of action to the fulcrum the 
weight-arm. 

54, Law for First Class. — In order that P and W may- 
just balance each other we must have, as can be seen from 
Exercise 8, 

power X power-arm = weight X weight-arm. 

Tins is the law for a lever of the first class. 

55. Levers of the Second and Third Classes. — But we 

may have a case, like that shown in Fig. 29, in which the 
line of action of the weight lies between the fulcrum and 
ihe line of action of the power. This arrangement gives us 
what is called a lever of the second class. 




Fig. 29. 



Fig. 30. 



There is still a different case, shown in Fig. 30, where 
the line of action of the power lies between the fulcrum 
and the line of action of the weight. This is called a lever 
of the third class. 



In the second and third classes of levers, as in the first 
class, the shortest distance from the fulcrum to the line of 



THE LEVER. 51 

action of the weight is called the weight-arm, and the 
shortest distance from the fulcrum to the line of action of 
the power is called the potver-arm. 

The pupil is to find out by means of the following Exer- 
cise whether the laws of the second and third classes of 
levers are as simple as the law of the first class. 

EXERCISE 10. 

LEVERS OF THE SECOXD AND THIRD CLASSES. 

Apparatus: The lever (No. 17) supported as in Exercise 8. A 
scale-pan (No. 18). A set of weights (No. 19). A spring-balance 
(No. 7). 

Suspend the pan with a load of 8 oz. at a point 5 cm. from the 
middle of the lever, and, on the same arm of the lever, at a distance 
of 10 cm. from the middle, pull upward with, a spring-balance, con- 
nected with the lever by means of a loop of thread, until the weight 
is balanced and the lever becomes horizontal. You have here a lever 
of the second class. Read the spring-balance and record as follows : 

Lever of Second Class. 

Weight. Weight-arm. Power. Power-arm. 

9 oz. 5 cm. .... 10 cm. 

Try other similar cases, and study them all until you are able to 
write down the law for this class of levers. 

Then with the same apparatus place the spring balance between 
the fulcrum and the line of the weight. You will now have a lever 
of the third class. Try various cases and record as before 

Lever of Third Class. 

Weight. Weight-arm. Power. Power-arm. 



Law. 



QUESTIONS. 

(1) A lever supported at its centre of gravity is used to lift a weight 
of 100 lbs. applied at a distance of 1 ft. from the fulcrum, The 



52 PHYSICS. 

power is applied 5 ft. from the fulcrum and on the opposite side from 
tbe weight. How great must the power be ? Must the power be 
applied upward or downward ? 

(2) If the power were placed on the same side of the fulcrum as 
the weight, everything else being as described in the preceding 
problem, how great would the power have to be ? Would it be 
applied upward or downward ? 

(3) If the power were 50 lbs. applied at a point 2 ft. toward the 
right from the fulcrum, and if the weight were applied 8 ft. toward 
the right from the fulcrum, how great could the weight be ? 

(4) If a weight of 5 lbs. were placed 4 ft. toward the right from the 
fulcrum, and a weight of 7 lbs. 6 ft. toward the right from the ful- 
crum, how far from the fulcrum toward the left must a force of 10 
lbs. be applied in order tD make the whole balance ? Arts. 6.2 ft. 

In the four preceding problems the weight of the lever has not been 
considered, because the centre of gravity has been supposed to be at 
the point of support. Suppose now that the lever weighs 4 lbs. and 
that its centre of gravity is 3 ft. to the right from the fulcrum, and 
with this new condition go over each of the four problems again. 

56. Force at the Fulcrum. — If we take a case like that 

shown in Fig. 31, it is plain 
that 4 oz. applied 7 cm. from 
the centre will balance 2 oz. 
applied 14 cm. from the cen- 
tre, but it may not be per- 
fectly plain how great the pull 
on the fulcrum itself is. We 
will, therefore, in the next 
FlG - si. Exercise try the experiment 

in one or two simple cases and see what the result will be. 

EXERCISE II. 

FORCE EXERTED AT THE FULCRUM OF A LEVER. 

Apparatus: The lever of No. 17 freed from its support. Two 
scale-pans (Nos. 18a and 18b). Two 1-oz. wts. and one 2-oz. wt. 
from No. 19. The spring-balance (No. 7). A piece of copper wire 
about 1 mm. in diameter bent into the form of a hook {h in Fig. 32). 
A piece of thread about 15 cm. long. 



ID 




o 



THE LEVEB. 53 

Suspend the bar from the balance in the manner indicated by Fig. 
32. Xote and record the weight 
of the bar alone. Then suspend 
one scale-pan with a 1-oz. weight 
from one arm of the bar, and the 
other scale-pan with a 2-oz. weight 
from the other arm in such a way 
as to balance, taking care not to let 

the pans and weights spill. Xote r 

and record the reading of the bal- 
ance. Then make the loads (pan FlG 32 
and weight) 2 oz. on one side and 4 

oz. on the other, and read and record. Try any other experiments 
that you can with the weights furnished, until you feel reasonably 
sure that you know the relation between the weights applied and the 
pull on the balance. Then state what this relation is. 

57. Laws of the Lever. — In each of the cases in Exercise 
11 we have applied two downward forces to the bar in sus- 
pending the two scale-pans with their loads, and have found 
these two forces to be balanced by another force exerted 
upward by the spring-balance. It will be well for us to 
study such cases very carefully, for similar ones are often 
found. 

Suppose we are to make three parallel forces, A y B, and 
(7, just balance each other when all are applied to the same 
body. Can we from what we have now learned tell any- 
thing about the relative magnitude and the arrangement of 
these forces ? 

We know that — 

1st. All the forces cannot point in the 
same direction. Let us suppose that G is 
opposite in direction to A and B. 

2d. Tlie force C must he equal to the sum 
of the two forces A and B. 
B\/ 3d. The line along which C is applied 
fig. 33. must lie between the lines along ichich A 

and B are applied. 



I 



51 



PHYSICS. 



4zth. A X shortest distance from line of A to line of 
C = B x shortest distance from line of B to line of C. (See 
Fig. 33.) 

These rules apply as well to horizontal forces as to ver- 
tical forces. Try three spring-balances laid parallel to each 
other on a table and pulling at some light horizontal bar — 
a lead-pencil, for instance. (Or try the " checker-board " 
with larger spring-balances, if the apparatus of the Second 
Part is available.) 

58. Pulleys. — We have already learned to consider a disk 
pivoted at the centre as a kind of lever. When such a lever 
is worked by means of a cord or band lying upon its cir- 
cumference, it is called a pulley. We shall now see that 
the pulley form of lever has some great advantages. 

EXPERIMENTS. 

(1) Take the p alley shown in Fig. 34 (No. XV), and let ns first use 
the largest circle only. 




Fig. 34. 

If we fasten two equal weights, W x and TT 2 , to the ends of a 
string and pass the string across the top of the pulley, we shall of 
course find that they balance each other. 



THE LEVER. 



55 



But suppose we used two strings, one for TFi and the other for 
TT 2 , fastening each string to a pin or tack at point A, but letting 
each string rest in the groove of the pulley, so that the final position 
of the two strings will be represented by Fig. 34. Will two equal 
weights balance each other under these conditions ? The question 
is quickly answered by trial, and by turning the pulley a little one 
way or the other we can try the experiment with A in a variety of 
positions. 

(2) Next try the effect of a horizontal pull, P, applied by a spring- 
balance at the top of the pulley to balance a weight W, as in Fig. 
35. (Remember that in this position the reading of the ordinary 
8-oz. balance is about J oz. less than the real force exerted by it, be- 




Fig. 35. 



cause the spring of the balance does not now support the weight of 
the hook and bar, which is about \ oz.). Find by experiment 
whether the force P must be greater or less than, or equal to, the 
direct pull of the weight W. 

(3) Balance a weight on one circle of the pulley by a weight on 
another circle, and find the simple relation which holds between the 
balancing weights and the radii of the circles. 



59. Advantages of a Pulley. — We see that the advantage 
of a pulley, as compared with a simple bar-lever, is that the 
pulley enables us to vary the direction of our power at will 




56 PHYSICS. 

and to lift a weight a much greater distance than we could 
with a bar-lever no longer than the diameter of the pulley. 
In fact, the distance through which we can lift the weight 
by means of the pulley depends merely upon the length of 
the string that supports the weight. 

60. Windlass, Capstan, etc. — The windlass (see Fig. 36) 
is a familiar apparatus consist- 
ing of an elongated pulley, d, 
called the drum or cylinder, 
turned by a power applied at 
the handle, A, and acting 
through the lever, or crank, c. 

The crank and handle are 
sometimes called a tvinch. The fig. 36. 

same name is sometimes applied to the whole windlass. 

If the power is applied at right angles with both h and £, 
and if the cord sustaining the weight is small, we have, 
very nearly, 

P X length ofc — W X radius of d. 

A capstan is the same in principle as a windlass, but has 
a vertical drum, so that the lever travels in a horizontal 
circle. 

On ship-board capstans are frequently worked by means 
of a number of men walking about the drum and pushing 
against the levers, or capstan-bars, of which there may be 
several applied to one cylinder. 

61. Movable Pulleys. — In the pulleys thus far studied 
by us the pivot has been fixed in position; but pulleys 
with movable pivots are frequently used. 

EXPERIMENTS. 

(1) Take the small metal pulley (No. XVIII) and arrange it ac- 
cording to the indications of Fig. 37, P being the pulley, M a weight 



THE LEVEE, 



Oi 




suspended from an axis through the centre of the pulley, B a spring 
balance, and h sl hook to which one end of the string 
passing beneath the pulley is attached. 

To what class of levers does the pulley in this 
position belong? What, then, should be the rela- 
tion between the weight, which is M plus the weight 
of the pulley itself, and the pull exerted by the 
spring-balance? Find by experiment whether the 
conclusion reached is correct.* 

(2) Let us now try an arrangement, like that 
shown in Fig. 38, in which we have one pulley, A, 
hooked to a bar overhead, and a double pulley 
(No, XIX), B, which moves up and 
down with the load M. 

Let us consider what should be 
the relation between the pull P and 
the weight W, which is M plus the 
weight of B, in this case. 

In the case tried in Experiment 1 
we had two strings holding up the 
pulley P. We have now four 
strings holding up the pulley B. 
After thinking upon the matter for 
a little time, trying to study out 
what is the relation between P and 
TFwith this arrangement, let us try 
the experiment as we have already 
tried it in the simpler case, noting 
the force shown by the spring-balance when M is moving steadily 
up, and again when it is moving steadily down, and taking the mean 
between these two forces as the one that would be required to bal- 
ance the weight, W, if there were no friction. 

* In making this trial one must remember that friction is often 
large in pulleys, even when they are well oiled, as this one should 
be. Now when the load is being steadily raised the hand carrying 
the spring-balance must lift harder than it would if there were no 
friction, but when the load is being steadily lowered, the hand, pull- 
ing just hard enough to prevent the load from, hurrying, is assisted 
by the friction. The mean between the reading of the balance going 
up and the reading of the balance coming down will show, very 
nearly, what the pull required to sustain the load would be if there 
were no friction. 



M 



Fig. 38. 



58 PHYSICS. 

62. Another Law for Relation of Power to Weight. — 

The law, "4th" in § 57, for the relation between potver and 
lueight is extremely useful, and, properly applied, is suffi- 
cient for very complicated cases of the lever and pulley ; but 
in some cases it is more convenient to make use of a differ- 
ent form or statement of this law, a form which makes use 
of the relative distances moved over by the power and the 
weight in any operation of the lever, or pulley, or combina- 
tion of these, that may be in action. 

63. Search for the Law. — If we study the various cases 
of lever, pulley, and combination of pulleys that have been 
described in the preceding pages, we shall see that when- 
ever the iveight is greater than the poiver the weight moves 
a less distance than the power does in any given operation 
of the apparatus; but whenever the weight is less than the 
power, friction being supposed zero, the weight moves a 
greater distance than the power does in the operation of 
the apparatus. 

64. Statement of the Law. — If we study the matter 
more closely, we shall find the following rule or law sug- 
gested, though we cannot say that it is proved by our pre- 
ceding experiments: 

PX D p = WxD w9 

where P stands for the power applied ; 
W " " " weight lifted; 
D p " " " distance the power moves ; 
D w " " " " " weight moves. 

APPLICATIONS OF THIS LAW. 

(1) It is evident that this law can be readily applied to a case like 
that of Fig. 38. We can see at once that if the pulley B were 
lifted one inch while P remained stationary there would be four 
inches of loose string under B. To make the string taut again, P 
would have to rise four inches. In actual use P and B rise at the 
same time, P moving four times as fast as B, 



TEE LEVER 59 

(2) In Fig. 39, representing the rear wheel and gear of a bicycle, let 
the diameter of the tire be 28 in. ; 
" " " " small sprocket-wheel be 2 \ in. ; 

" " " " large sprocket-wheel be 5 in. ; 

" length of pedal radius be 6 in. 




Fig. 39. 

If the weight of the rider, 150 lbs., rests entirely upon the pedal 
shown, in its present position, how great a weight acting in opposi- 
tion, as in the figure, will just neutralize the driving-power ? 

The circumference of the circle described by the pedal is (see Ex. 
B) 2ft X 6 inches, or 12;r inches. 

One revolution of the pedal-crank makes the tire revolve 2 times, 
which drives the bicycle forward 

2 X 27T X 14 inches = 567T inches. 

The weight W will be lifted as fast as the bicycle moves forward. 

If Dp stands for the distance the power P moves downward from 
its present position during any very short time, and if D w stands 
for the distance W moves upward during the same time, we have 

Dp \D lc \ :12tt :56tt. 

Hence the law W X D w = P X D p (see § 64) gives 

WX 56/r = PX 12k, 
or W=j\P= T 3 T X 150 = 32. 1 + (lbs.) 

QUESTIONS. 

(1) A large pair of shears is used to cut a wire. One handle of the 
shears being held fixed, a power of 25 lbs. is applied to the other 
handle at a distance of 2 ft. from the pivot. The wire cut is placed 
2 in. from the pivot. How great is the resistance offered by the 



60 PHYSICS. 

wiie ; that is, bow great a force applied just over tlie wire would 
drive the blade through it ? 

(2) A man lifts 10 lbs. of coal on a shovel. His left hand is at the 
end of the handle ; his right hand is 18 in. distant from his left hand ; 
the centre of gravity of the coal is 36 in. distant from the left hand. 
The shovel itself weighs 6 lbs. and its centre of gravity is 21 in. from 
the left hand. 

(a) What is the direction and magnitude of the force exerted by 
the left hand ? (Consider the right hand as the fulcrum.) 

Ans. 11 lbs. 

(&) What is the direction and magnitude of the force exerted by the 
right hand ? (Answer this question as if the left hand were replaced 
by a weight.) 

(3) A body weighing 160 lbs. is suspended from a pole resting on 
the shoulders of two men, A and B, of equal height. The point of 
suspension is 3 ft. from A's shoulder and 5 ft. from B's. How 
much of the weight does each man bear ? 

(4) Six men are working at a capstan, each exerting a force of 20 
lbs., each at a distance of 6 ft. from the centre. The diameter of 
the coils in which the rope is being wound on the drum is 1 ft. 
How great is the strain on the rope, all friction being neglected ? 

(5) A man finds that by moving one point of a machine, consisting 
of levers and pulleys, forward 10 in. he can move another point of 
the machine 1 in. If a force of 5 lbs. is applied at the first point, 
how great a resistance applied at the second point will be required to 
neutralize it, if there is no friction in the machinery ? 

(6) Can the class name any tools or machines, not already men- 
tioned in this book, in which levers or pulleys are used ? 



CHAPTER V. 

THREE FORCES DIRECTED THROUGH ONE POINT: THE 
PARALLELOGRAM OF FORCES. 

65. Introductory. — In studying the lever we have 
usually, though not always, had parallel forces to deal with, 
forces acting straight up or straight down. But very often 
we have to do with bodies that are acted upon by forces not 
parallel to each other. Thus when a ladder standing upon 
the ground leans against a house we have at least three 
forces acting upon the ladder: 1st, the earth's attraction, 
or, as we call it often, the tv eight of the body, which acts 
as if the whole substance were at the centre of gravity; 2d, 
the push of the ground against the foot of the ladder, 
which push is not straight upward; 3d, the push of the 
wall against the top of the ladder. 

Again, a flying kite is acted upon by the earth's pull, 
straight downward ; by the force exerted by the air, which 
force, because of the wind, is not straight upward ; by the 
pull of the string, which pull is not straight downward. 

The way to begin the study of such cases is to study the 
case of three forces all acting straight from or straight 
toward a single point. We shall take such a case in Exer- 
cise 13, measuring the forces by means of spring-balances. 

66. Errors of Spring-balances. — It is quite possible in 
specific-gravity work to get accurate results with an in- 
accurate balance ; for the specific gravity of a body is found 
by taking the ratio of two quantities, both found by weigh- 
ing with the same balance, and if the balance should give 

61 



62 PHYSICS. 

the weight of each as n times its true value the ratio of the 
two false weights would be the same as the ratio of the true 
weights, whatever the value of n. 

But though a balance may give the weight of everything 
as n times its true weight, n being the same for all parts of 
the scale, it is very unlikely that several balances will all be 
wrong in just the same way; and as in Exercise 13 we 
shall need to use three balances in combination, it is neces- 
sary for us to give more careful attention to their errors 
than we have given heretofore. 

The following Exercise is intended to show how the errors 
of a spring-balance may be found and may be recorded in a 
form convenient for future use. 

EXERCISE 12. 

ERRORS OF A SPRING-BALANCE. 
Apparatus : A spring-balance (No. 7). A set of weights (No. 19). 
Thread for suspending the weights. A measuring-stick (No. 3). 
(Although weights reckoned in ounces are referred to in this Exer- 
cise, it may be performed equally well with suitable weights reckoned 
in grams.) 

Balance in the Vertical Position. — Suspend the balance by 
its ring from some convenient support so that the index will be not 
higher than the eye of the observer. 

Make five careful readings with the loads indicated in the first 
column below, and record these readings in a second column ; thus, 
for example: 

True Load. Reading. Error.* 

oz. - 0.2 oz. - 0.2 

2 " 2.0 " 0.0 

4 " 4.2 " +0 2 

6 " 6.3 " +0.3 

8 " 8.1 " +0.1 




* That is, the quantity which must be subtracted from the reading 
in order to find the true weight. 



THE PARALLELOGRAM OF FORGES. 68 

Now draw in the notebook a straight line a little more than twice the 
length of the scale of the balance, and mark off on it points corre- 
sponding to the loads and readings given above. 

Then represent the errors by vertical distances, measured down 
from the points indicating the readings when the errors are negative, 
and up when they are positive, showing these errors on a rather 
large scale, 0.5 cm. per 0.1 oz., for instance. 

Draw a curve through the extremities of these vertical distances, as 
in Fig. 40. This curve will enable us to tell with sufficient accu- 
racy for our present purposes the errors of any other readings made 
with the balance in the vertical position ; for instance, if the balance 
to which Fig. 40 relates reads 1 oz., we may assume that the error is 
very nearly 0.1 oz. and that the true weight is very nearly 1.1 oz.; if 
the reading is 3 oz , we may assume that the true weight is about 2.9 
oz., and so on, the error for each case being found by measuring 
from the point indicating the reading up to or down to the curve. 

The Same Balance in the Horizontal Position. — We will 
suppose at first that the same index is used for the horizontal as for 
the vertical readings. 

Lay the balance flat on its back and tap it gently several times.* 
Then take its reading, which we will suppose to be — 0.5 oz. In this 
case it would require a force of 0.3 oz. to pull the index down to the 
position it occupied with no load in the vertical test. To bring the 
index to any reading in the horizontal use of this balance will, there- 
fore, require a force 0.3 oz. greater than the weight which, applied 
to the hook in the vertical use of the balance, would bring the index 
to the same point. 

It would in the case here supposed be sufficiently accurate for our 
purposes to add 0.3 oz. to any horizontal reading, and then correct 
this increased reading by means of the curve given in Fig. 40. 

When a different index is used for horizontal readings, the princi- 
ple is the same. For example, let us suppose the second index reads 
— 0.1 without pull in the horizontal use of the balance. We say, 
this reading exceeds the vertical reading without load by 0.1 oz. We 
must, therefore, subtract 0.1 from all horizontal readings made with 
this second index, and then correct these reduced readings by means 
of the curve in Fig. 40. 

* This method assumes that the movable parts of the balance are 
restrained by friction only, which the tapping overcomes. Sometimes 
the slot in the balance-face is not long enough to allow the index to 
find its unrestrained position. 



64 



PHYSICS. 



The curve so often referred to should be marked with the number 
of the balance and kept for future use. 

EXERCISE 13. 

THREE FORCES IN ONE PLANE AND ALL APPLIED AT ONE 
POINT: PARALLELOGRAM OF FORCES. 

Apparatus: Three 8-oz. spring-balances, each provided with two 
small blocks (No. 22) to go under its sides and hold it flat on its back 
when it is lying upon the table. The rectangular block (No. 9). 
The measuring-stick (No. 3). A sheet of paper. Thread. 

Take two pieces of strong thread, one about 12 inches, the other 
about 6 inches, long, and tie one end of the short thread to the middle 
of the long one. Fasten the three loose ends to the hooks of the 
spring-balances; then lay the latter upon the table, putting the blocks 
under their sides, as in Fig. 41, and let one student pull at each bal- 




Fig. 41. 
ance, taking care that the slit of each balance- face is in a straight line 
with the thread, until no one of these reads less than 3 oz. 



THE PARALLELOGRAM OF FORGES. 



65 



It will be found that any variation in the angles which the strings 
make with each other will require a change in the forces. Evidently 
there is some connection between the directions of the strings and the 
forces necessary to balance each other. The object of this Exercise 
is to make out what this connection is. 

Put under the threads a sheet of paper, and draw on this paper, 
just under each thread, apencil-mark parallel to the thread, and then 
write down alongside each pencil- mark the force in the direction of 
that line, as shown by the spring-balance. The balances must be 
held very still while these lines are being drawn, and must be read 
before any change occurs in the direction of the lines.* To draw a 
line place one side of the block (No. 9) close alongside one branch of 
the thread, taking care not to push the thread out of place, and then 
run the point of a well-sharpened pencil along the edge of the block 
under the thread. Draw the other lines in the same way, doing all 
very carefully. 

Each student in turn should make a set of lines, and record along- 




Fig. 42. 

side them the proper forces. The directions of the pulls should be 
varied somewhat by each experimenter, in order that his lines and 
forces may not be exactly like those of others. 

Take now the wooden ruler (Xo 3), and extend the three lines 
toward each other till they meet at one point. This they will do if 

* It is well to fasten the ring of each balance to some object heavy 
enough to hold the balance in place, thus relieving the experimenters, 
who might grow tired and unsteady in holding the balances long 
enough to permit of drawing the pencil- marks properly. 



66 



PHYSICS. 



they have been drawn originally just under the threads. If they do 
not all meet at one point, a new line should be drawn parallel to one 
of them in such a position as to pass through the crossing of the other 
two lines, and this new line, the dotted line in Fig. 42, is then to be 
used in place of the original line. The three lines as now drawn will 
represent accurately the directions of the three forces. 

Now measure off from the common point along the line A a dis- 
tance of 1 cm. for each ounce (or each 30 gm., if the forces are meas- 
ured in grams) of the force which was exerted along that line, and 
put a small arrow-head (see Fig. 42) at the end of this measured dis- 
tance. Erase that part of line A which lies beyond the arrow-head. 

Do the same with lines B and C that has been done with A. The 
three arrows thus obtained, all reaching from the same point, repre- 
sent the magnitude and the direction of the three forces exerted by 
the spring-balances. 

Now with A and B of Fig. 42 as two of the sides draw a parallelo- 
gram, taking pains to make it accurate.* Then make a parallelogram 
with B and C as sides, then one with A and G as sides. Compare the 

* One line may be drawn very nearly parallel to another by means 
of a device illustrated by Fig. 43. LI is a line already drawn. The 
block (No. 9) is so placed that for an eye placed at E the edge mn 

E 




Fig. 43. 



appears to be close to LI and parallel to it. Then a pencil-mark is 
made along the edge op. 

A better method is to set the edge op on the line LI and then guide 
the block to a new position by sliding it along the straight edge of a 
ruler at right angles with LI. 



THE PARALLELOGRAM OF FORCES. 



67 



length and direction of the line C with the length and direction of 
the diagonal of the parallelogram AB\ the line A with the diagonal 
of the parallelogram BC; the line B with the diagonal of AG. 

From a study of the Exercise make a rule showing how to find 
the direct. on and magnitude of a force G which put with two forces 
represented by the lines A and B (Fig. 44, below) will just balance 
them. 

Applications of the Parallelogram of Forces. 

The rale found in the preceding Exercise, and which is called the 
'parallelogram of forces, is peculiarly easy to apply in cases where 
two of the forces are at right angles with each other. A number of 
such cases will now be discussed. 

(1) A force of 7 lbs. pulls north from a certain point, and a force 
of 4 lbs. pulls east from the same point. How large must a third 
force be to hold them in check, and what will be its general direc- 
tion? 

Solution. — Fig. 45 indicates the method of working the problem. 







Fig. 44. Fig. 45. Fig. 46. 

The directionoi the third force is evidently southwest, in a line with 
the diagonal of the rectangle, of which the base is 4 and the height 7. 
The magnitude of the third force is evidently equal to the length of 
the diagonal, which a simple rule of geometry shows to be 

v 4 -2 + r - = 8 - 06 +• 

(2) A boy weighing 50 lbs., represented by the point b in Fig. 46, is 
seated in a swing 10 it. long, represented by the line 8b. A horizon- 



68 



PHYSICS. 



tal pull holds him 4 ft. to one side from the natural position of the 
swing. 

(a) How great is the pull of the swing- rope ? 

(b) How great is the horizontal pull ? 

Solution. — Represent the weight of the boy by the line b W, made 
1 cm. long, we will suppose. The other two forces acting on b must 
be such as to make a balance with this force. Draw the line bW 
exactly equal and opposite to bW, and complete the parallelogram 
as in the figure. 

A little geometry shows that the triangle MW is similar to the tri- 
angle Sbn. 



The line bi represents the pull of the swing-rope, 
triangles just mentioned are similar, we have 

U-.bW ::Sb : Sn, 



and, as the 



bi=(Sb-i-Sn) XbW 
= (10-*- VW^~4: 2 ) xbW'=lMxbW 
But b W = b W, which represents a force of 50 lbs. Hence bi rep- 
resents 54.5 lbs. Ans. to {a) = 54.5 lbs. 
The pull upon the rope is therefore greater than the boy's weight. 

The line bh y which is = i W, r epresents the horizontal pull. 
We have 

bh'.bW : :nb:nS, 



Ans. to (b) = 21.8 lbs. 



bh = (4 -r- VW -4?)XbW = 0.436 X b"W\ 
The force represented by bh is therefore 
0.436 X 50 lbs. =21,8 lbs. 
(3) A mass of 20 lbs. is suspended from 
the point p (Fig. 47), where a string is bent 
at a right angle. The ends of the string 
are fastened to two nails , i^i and i\T 2t 
which are at the same height. The part 
ATi^ is 4 ft. long, the part N^p is 8 ft. 
long. 

(a) How great is the pull upon iV a ? Fig. 47. 

(b) How great is the pull upon JV 2 ? 

Solution.— Represent the weight by pW. Draw p W equal and 
opposite to pW. Complete the parallelogram. 




THE PARALLELOGRAM OF FORCES. 69 

The triangle pqW is equal to the triangle W'rp and similar to the 
triangle N*pNi. The side N^N* = V& _j_ 42 = 8.94 ft. nearly. 

The pull on i\^ is represented by pq, and we have 

pq : p W : : N& : JMfi, or pg = (8 ■*- 8.94) X p IT'. 

Therefore pq represents very nearly 0.895 X 20 lbs. = 17.90 lbs., 
which is the answer to question (a). 

Similarly we find the pull on Jjf % to be very nearly (4 ■+■ 8.94) X 20 
lbs. = 8.95 lbs., which is the answer to question (5). 

The pull along the 8-ft. part of the string is just one-half as great 
as that along the 4-ft. part. 

The LtfCLEN'ED Plane. 

67. Introductory. — The parallelogram of forces will 
enable us to understand a contrivance very often used for 
raising heavy weights. It is a common thing to see barrels 
of flour or other heavy objects loaded upon wagons by roll- 
ing them up a plank or a pair of rails, placed with one end 
on the ground and the other upon the wagon, so as to make 
the ascent gradual instead of straight up. The flat slanting 
surface up which the body is rolled is called an inclined 
plane. 

Sometimes a body is lifted by forcing an inclined plane, 
the slanting face of a wedge , under it, as in Fig. 48. 




Fig. 48. 



Sometimes the force used by an experimenter or a work- 
man with the inclined plane is parallel to the inclined sur- 
face; sometimes it is parallel to the base-line of the plane, 
the horizontal surface of a wedge, for example. 

We will consider each case in order, seeking for the con- 



70 



PHYSICS. 



nection between the weight, steepness of incline, and force 
to be applied. 

68. Force Applied Parallel to Incline. — This case is illus- 
trated by Fig. 40, where 

L reDresents the length of the incline; 

B " " base " " 

H " " height" " 

W " weight of the body on the incline, 

applied straight downward from the centre of 

gravity of the body; 
W is the equal and opposite of IF; 
N represents the force exerted upon the body by the 

plane X, a force which is straight outward 

from the surface of the incline if there is no 

friction (see Chap. VI) between the body and 

the incline; 
P represents the pull, parallel to the plane L, which 

with the force N will just balance W. 




By comparing the dotted triangle with the triangle whose 
sides are L, B, and H we see that 

P : W (or IF) : : H: L, 



THE PARALLELOGRAM OF FORGES. 



71 



or 

PXL=WX H, 

P= Wx (H+L). 

EXPERIMENTS. 

Take apparatus No. XX and adjust it as indicated by Fig. 50, put- 
ting 7 oz. upon the pan, so that P= 7+ 1 = 8 oz. Then raise or 




Fig. 50. 

lower the incline till the weight W will barely roll up the incline 
when the apparatus is purposely jarred slightly. (The incline cannot 
be quite so steep when this takes place as it might be if there were 
no friction. If a knot is made in the thread near where it passes over 
the pulley at the top of the incline, a very slight movement up or 
down the incline can be detected by watching the position of this 
knot. A slight movement is enough.) 

As soon as this adjustment is made read H, the length of the ver- 
tical scale from the top of the base-board to the under side of the in- 
cline, and record in the way indicated in the table below (upper row 
of numbers). 

Then without changing P rase the incline somewhat more, until W 
will, when the apparatus is jarred, barely roll down the incline. (The 
incline must be somewhat steeper for this than it would have to be 
if there were no friction.) When the proper adjustment is made, read 
the new value of //and record it in the second line of the table below. 

To find the //that would make P just balance W if there were no 
friction, take the mean between the two values now recorded. Then 



72 



PHYSICS. 



find the L that would correspond to this value of H, L being the dis- 
tance along the inclined scale from the hinge to the point of crossing 
the vertical scale. 

P W H L 

Going up. ..8 oz. 16 oz. .... .... 

" down.. 8 " 16 " 



To balance ... 8 oz. 



16 oz. 



If time permits, make P = 6 oz., then 4 oz. ; and in each case repeat 
the operations just described. 

69. Force Applied Parallel to Base. — This case is illus- 
trated by Fig. 51. 




Fig. 51. 

The line W is not here, as it is in Fig. 49, the hypothe- 
neuse of the dotted triangle; but it is evident that the 
dotted triangle is similar to the triangle made up of X, B, 
and H. P is the force applied parallel to the base, and just 
sufficient, with N> to balance W. We have, from a com- 
parison of the triangles, 

P : W (or W) ::I1:B, 
or 

P X B= W X H 9 

P= WX (H+B). 



THE PARALLELOGRAM OF FORCES. 



73 



EXPERIMENT. 

For experiments in which the power is applied parallel to the base- 
line we cannot well make use of a string running over a pulley. We 
must apply the power by means of the spring-balance, as shown in 
Fig. 52, the long slot cut through the incline lengthwise allowing us 
to do so. 




Fig. 52. 

Find by trial a steepness of incline that will make P about 7 oz. , 
and, keeping this steepness unchanged for the time, find how large 
P is when it is pulling W slowly and steadily up the incline, and how 
large when it is letting TFrun with equal slowness and steadiness 
down the incline. Take the mean* of these two values as the one 
that would be needed to balance W if there were no friction. 



We record, then, for this case : 



Going up... . 
11 down. 



w 

16 
16 



To balance. . . 



16 



where B is the length of the base-line from the hinge to the foot of 
the vertical line, along which IL is measured. 

If time permits, lower the incline and try various degrees of steep- 

*The mean of the two values of P is not, in this case, exactly the 
quantity wanted, because the greater pull of P when IF is going up 
the incline makes W press harder against the incline when going up 
than when going down, thus increasing friction. The mean value of 
P. as now found, is a little greater than the value wanted, but so 
little that the error is not important, 



74 PHYSICS. 

ness, so that P will be in one case about 5 oz. and in another case 
about 3 oz. 

70. The Wedge. — The ivedge, as commonly used (see 
Fig. 48), is a case of the inclined plane with the applied 
force parallel to the base. It differs from the case shown 
in Fig. 51 in one respect. In Fig. 51 the body raised has 
a motion parallel to the base of the plane, while the plane 
itself has not. A wedge commonly has a motion parallel to 
its own base, while the body raised or otherwise moved by 
it does not have a motion in this direction. The principle 
involved in the two cases is quite the same, and for a wedge 
used to lift a weight we have, leaving friction out of ac- 
count as before, 

P=WX (H+B), 

where H stands for the thickness of the wedge, and B for 
its length. 

71. The Screw. — The screw is an ingenious form of the 
inclined plane, as the following experiment will show. 




EXPERIMENT. 

Cut out a long narrow triangle of paper, (see Fig. 53), and then 
wind it upon a lead-pencil, beginning at the end £Tand keeping the 
line B all the time at right angles with the length of the pencil. The 
line L will make a regular spiral around the pencil, corresponding to 
the thread of a screw. 

72. Pitch of a Screw. — The distance from one turn of 
the thread to the next turn, measured parallel to the length 
of the screw, is called the pitch of the screw. 



THE PARALLELOGRAM OF FORGES. 75 

It is evident that in the ordinary use of a screw one 
revolution moves it toward or backward the length of its 
pitch. 

73. Use of the Screw in Lifting. — A very large iron 
screw, called a jack-screw, is frequently used for lifting 
very heavy bodies. The power is applied to such a screw 
by means of a long handle or lever, which projects from the 
head of the screw at right angles with its axis, its central 
lengthwise line. Leaving friction out of account, we can 
find the relation between power applied and weight of the 
body lifted thus : 

Let P = the power applied to the handle at right angles 
with the handle and with the axis of the 
screw ; 
A = the distance from the point of application of P 

to the axis of the screw ; 
r == the radius of the screw itself; 
p = the pitch of the screw ; 
W= the weight of the body lifted. 
The force P produces at the thread of the screw a force 
P' = P x (A -f- r). (See Exercise 10.) 

This force at the thread is like the power used to drive a 
wedge. The circumference of the screw at the thread, 
which = 27rr, corresponds to the length of the base of the 
screw, while the pitch corresponds to the thickness of the 
wedge. We have, then 

P' X 27rr= Wxp, 

P'=z Wx (p + 2*r), 

or 

Px(iv r) = W X (p + 27tr) ; 
whence 

P X 2?rA = WXp. 

Observe that we have here, as we have had so often 



76 PHYSICS. 

before, the rule, Power X distance the power moves = weight 
X distance the weight is lifted. 

Definitions. 

74. Equilibrant. — A single force that will just balance, 
or make equilibrium with, two or more others is called their 
equilibrant. 

In Fig. 46 b IF is the equilibrant of hi and bh; 
U " " " " bW" bh; 

bh " " " " bW" bi. 

75. Resultant. — A single force that can exactly replace 
two or more others, so as to produce the same effect upon 
the body acted on, is called their resultant. 

In Fig. 46 bW is the resultant of bi and bh. 

The resultant and equilibrant in any given case are equal 
in magnitude, but opposite in direction, so that the two 
would exactly balance each other. 

QUESTIONS. 

(1) What is tlie resultant of pq and pr in Fig. 47 ? What is their 
equilibrant ? 

(2) Draw three lines leading from one point, giving to them such 
magnitudes and directions that they will represent three forces in 
equilibrium with each other. 

(3) Replace two of the lines in the preceding problem by two 
others that will also represent equilibrium with the third line. 

(4) A telegraph-wire pulls north from a post with a force of 12 lbs. ; 
another pulls west from the same post with a force of 16 lbs. 

(a) How great is the resultant pull on the post? 

(b) If a third wire is put in to neutralize the pull of the other two, 
should it pull more nearly south than east, or more nearly east 
than south ? 

(5) Two sticks of equal length, OA and OB in Fig. 54, each rest 
ing one end upon the ground, meet at a right angle in a frictionless 
joint 0. From this joint is suspended a mass of 5 lbs., the weight 
of which is represented by the line OW. 




THE PARALLELOGRAM OF FORCES. 11 

(a) How great is tlie total force exerted by each stick against the 
ground, the weight of the stick being 
left out of account ? Ans. 3.54 — lbs. 

(6) How great is the vertical push ex- 
erted by each stick against the ground? 
Ans. 2.5 lbs. 

(c) How great is the horizontal push 
exerted by each stick against the FlG - 54 - 

ground? Ans. 2.5 lbs. 

(We see from this problem that the total vertical force exerted at 
A and B is just equal to the weight of the suspended mass, the 
weight of the sticks not being considered. If we think of AOB as 
one end of the roof of a house, the answer to (c) shows the tendency 
of the roof to push the walls apart. This tendency is met in actual 
roofs by beams or rods connecting A and B.) 

(6) A wedge 1 ft. long ou its base and 2 in. thick is used to lift a 
weight of 300 lbs. in a case where friction may be left out of ac- 
count. How great is the force, parallel to the base, required to 
drive the wedge ? 

(Friction not being considered, the force required to keep the 
wedge moving after it is started is no greater than the force required 
to hold it in place so as to make equilibrium, as in the discussion of 
§70.) 

(7) A safe weighing 2000 lbs. is resting on an inclined plane 12 ft. 
long, one end of which is 2 ft. higher than the other. How great is 
the force, parallel to the incline, required to keep it from sliding 
down ? 

(8) A jack-screw having a pitch of £ in. and a handle 2 ft. 1 in. 
long is used to lift a mass of 5000 lbs. How great must be the power 
applied to the end of the handle ? 



CHAPTER VI. 

FRICTION. 

76. Introductory. — When we push a heavy block along 
on the top of a table we feel a certain resistance. We 
know from experience that by making the surface of the 
table and the surface of the block yery smooth we can lessen 
the resistance. This resistance, the amount of which 
depends upon the condition of the rubbing surfaces, is ccdled 
Friction. 

Friction always opposes motion, whatever may be the 
direction of the motion, that is, it merely tends to stop 
the motion. It never helps to push the block back to the 
position where it started. 

We shall in Exercise 14 measure in a number of cases the 
force required to keep a block moving steadily along on a 
sheet of paper laid upon a level board, and shall study these 
cases with the purpose of finding out some useful facts, or 
laws, concerning friction between solid bodies. 

EXERCISE 14. 

FRICTION BETWEEN SOLID BODIES. 

Apparatus : A spring-balance (No. 7). A rectangular block (No. 
9). Set of weights (No. 19). A smooth sheet of paper about 1 ft. 
wide and 1^ ft. long. Thread. 

We shall first consider the velocity of the motion, that is, ice shall 
ask whether the force required to keep up a slow steady motion is 
greater or less than that required to keep up a more rapid steady 
motion. 

78 



FRICTION-. 79 

Lay the block on one of its broad sides, and attach it to the spring- 
balance by a thread passing around but not under the block. Load 
the block with weights until the force required to maintain a slow 
steady motion is about 3 oz. Draw the block parallel to its grain 
along the sheet of paper several times with a very slow steady 
motion, and then several times with an equally steady motion two or 
three times as fast. (As the paper is likely to grow somewhat 
smoother under the repeated rubbing, the experimenter should not 
make all his slow trials first, but should change from slow to fast 
and fast to slow a number of times.) 

Record your conclusion as to whether the slow or the more rapid 
motion requires the greater force. 

We shall next try to find out whether, the total weight being the 
same as before, it is easier or harder to draw the block on a narrow 
side than on a broad side. 

Use the same block and the same load of weights, pulling it now, 
as before, parallel to its grain. 

(The side upon which the block slides should in all cases be clean, 
and the broad and narrow sides which are compared should be, as 
nearly as practicable, equally smooth. The thread must not be 
between the rubbing surfaces in any case.) 

Record your conclusion as to whether the broad side or the narrow 
side offers the greater resistance to the motion. 

Finally y we shall ask what connection there is between the total mass 
drawn and the force required to draw it. 

For this purpose vary the weights placed upon the block, using 
not less than 6 oz. for the least and as much as 16 oz. for the greatest 
load. 

Add to the load in each case the weight of the block itself, and 
make the record in the following form, W being the load and b the 
weight of the block : 

W -f b* F (Force Required). 



Look for any simple relation between (1F+ b) and F. 

* It is well to begin with the lightest load, proceed in regular 
order to the heaviest, then go back in exactly the reverse order to 
the lightest, recording both trials made with each load and taking 
the mean of the two for final study. 



80 PHYSIOS. 

The experiments just described will teach a number or useful facts 
about friction between two solid substances, but one must be careful 
not to apply the conclusions here arrived at to extreme cases, ex- 
tremely slow or very fast motion, for id stance ; or to cases where the 
pressure is great enough and the edge of the sliding body narrow 
enough to cause an actual cutting of the body into the surface over 
which it should slide. 

77. "Laws of Friction." — The so-called laws of friction 
between solids are: 

1st. Friction is independent of the velocity of one surface 
across the other, other things being equal. 

2d. Friction is independent of the area of the rubbing 
surfaces i other things leing equal. 

3d. Friction is proportional to the total pressure of one 
surface against the other, other things ieing equal. 

The experimenter in Exercise 14 need not be surprised 
or disappointed if his observations do not agree exactly 
with these statements. In fact, the " laws " are not strictly 
true; but they are near enough to the truth to be of very 
great use. 

78. Coefficient of Friction. — If the pressure between two 
surfaces, at right angles ivith each of 'them, is called P, and 
if the friction between the two surfaces is called F, the ratio 
F-+- P is called the coefficient of friction. 

In Exercise 14 the ratio F -^ (TF+ b) is the coefficient 
of friction. 

There is a method of finding this coefficient without 
measuring either P or F. It makes use of an inclined 
plane and the parallelogram of forces. 

In Fig. 55 is supposed to be a body resting upon the 
incline AB, which is just steep enough to keep moving 
with uniform velocity, in spite of friction, if it is once 
started down the incline. 

The line W represents the weight of the body. This 



FRICTION, 



81 



is equivalent to a force OP at right angles with the incline 
and a force OM down the incline. It is the force OP that 




Fig. 55. 



causes the friction. It is the force OM that maintains 
motion in spite of the friction. 

If the body moves with uniform velocity down the 
incline, as we have supposed, the force OM must be exactly 
equal and opposite to the resistance of friction. For if OM 
were greater than the friction, the body would move faster 
and faster down the incline; while if OM were less than 
the friction, the body would move more and more slowly 
down the incline.* While the body is moving downward 
friction is represented by the arrow pointing from toward 
A, equal and opposite to OM. 

Therefore, in accordance with the definition given at the 
beginning of this article, we have 

coefficient of friction = OM -f- OP. 



* It require- force to set any body in motion and it requires force 
to stop any body that is in motion. If a body is moving along in a 
straight line with uniform velocity we know that the various forces 
acting on it balance each other. This matter is discussed further in 
the Second Part. 



82 PHYSICS. 

A comparison of the triangle OWM, in which WM = 
OP, with the triangle ABC shows that they are similar, 
and hence 

OM: OP : : AC: PC. 

That is, the 

coefficient of friction — AC -^ PC. 

EXERCISE 15. 

COEFFICIENT OF FRICTION. 

Apparatus : The same block tliat was used in Exercise 14. A 
flat board (No. 20) about 15 cm. wide and 50 cm. long for the block 
to slide on. A sheet of paper to cover one side of this board. Some 
means of raising one end of the board and adjusting it so that the 
block will just slide down it ; another block similar to the one which 
slides, or any similar object, will do for this purpose. A 30-cm. 
measur i ng-stick . 

Place one end of the board on the table and the other on the sup- 
port. Vary the steepness of the board by varying the position of 
the support, until such an inclination is found that the block, once 
started slowly, will barely continue in motion down the board. 

Then lay off on the table beneath the board a distance of 30 cm., 
measured from the edge where the board rests upon the table, and 
from the end of this line measure H, the vertical distance up to the 
under side of the board. The coefficient of friction will be H^- 30. 

If the same block, the same side of the bloc'v, and the same kind 
of paper are used in this Exercise as in Exercise 14, the value of the 
coefficients obtained in the two Exercises should be compared. 

79. Friction in Applied Mechanics. — Friction is one of 
the most important conditions in the construction and 
operation of very many mechanical appliances. It enters 
largely into the list of resistances to be overcome, as in the 
rolling friction of the car-wheels upon the track or of 
wagon-wheels upon common roads. Every axle revolves in 
its bearings with a measurable amount of friction, which 
can be diminished but not overcome by oiling the surfaces 



FRICTION. 83 

in contact. On the other hand, many machines and 
mechanical appliances would be valueless without friction. 
Upon this the efficiency of belting, of brakes, of nails and 
screws of every description, is dependent. The driving- 
wheels of engines or of electric street-cars, the feet of men 
or of horses, would be unable to produce or maintain loco- 
motion without the aid of friction. If its operation were 
suspended, every river would become a cataract, soon run- 
ning itself out. 

Rolling Friction. 

80. Introductory. — The friction encountered by a mov- 
ing body is usually much less when it is on wheels or rollers 
than when it slides, though it is true that on snow runners 
are better than wheels. The wheels of ordinary carriages 
do not get rid of sliding friction altogether, for the surfaces 
of the axle and the hub slide over each other. The " ball 
bearings " of bicycles do away with this sliding friction 
almost completely. 

81. Coefficient. — The coefficient of rolling friction of 
iron wheels on iron rails may be as small as .002,* so that 
a pull of 4 lbs. may keep in motion a carriage weighing 
2000 lbs. 

The coefficient of sliding friction of smooth dry iron upon 
iron is perhaps .15 or .20. 

82. Slipping of Wheels. — When people were first con- 
sidering the use of steam for dragging railroad trains, they 
thought it would be necessary to provide the driving-wheels 
of the locomotive with cogs fitted to a cogged rail along the 
track. This device was found to be unnecessary for ordi- 
nary work, but it is used on very steep inclines running up 
the sides of mountains. 

* Rankine, Civil Engineering. 



84 PHYSICS. 

Even upon ordinary railroads, when the rails are wet and 
there is a heavy train to be set in motion, the driving- 
wheels sometimes slip and revolve, while the train refuses 
to start. The frequency of the puffs from a locomotive 
depends upon the speed of revolution of the driving-wheels, 
and when an engine that has been p tiffing very slowly in 
starting a train suddenly gives three or four puffs in very 
quick succession, we may conclude that the driving-wheels 
are slipping on the rails. Engines are provided with sand- 
boxes, from which sand can be sprinkled upon the rails in 
front of the driving-wheels when slipping occurs. 

Friction between Solids and Fluids. 

83. Unlike Friction between Solids. — The laws of fric- 
tion between solids and fluids are very different from those 
which hold between solids. Friction between solids and 
fluids changes comparatively little with change of pressure, 
but it changes a good deal with change of velocity. The 
resistance of the air is an important obstacle to rapid 
motion, as in the case of a railroad train, and the frictional 
resistance of the water to the hull and propeller of a steamer 
demands most of the steam-power required to propel the 
vessel. 

84. Friction in Tubes. — The friction of liquids or gases 
flowing rapidly through long tubes is very considerable, 
as the following experiments will show. 

EXPERIMENTS. 

(1) Take a rubber tube 2 or 3 in. long and about 0.6 cm. in diame- 
ter of bore. Cue off a piece about 20 cm. long. Fill a large glass 
■Jar with water. 

Using the short piece as a siphon, keeping the lower end about 
10 cm. beneath the surface of the water in the jar, find the number 
of seconds required to fill a small tumbler with the water delivered, 



FRICTION, 85 

Try the same experiment with the long tube, keeping its outlet 
also 10 cm. below the surface of the water in the jar. 

Compare the rates of delivery in these two cases. 

(2) Take a rubber tube 2 or 3 m. long and about 0.15 cm. in diame- 
ter of bore. Cut off a piece 10 cm. long. Light a candle. 

Put out the candle-flame by Wowing through the short tube. See 
how far from the outlet of the tube the flame must be placed in 
order to survive the blowing. 

Repeat the trial, using now the long tube. 

Compare the distances in the two cases. 

Something more concerning friction of water in tubes is 
given in the Second Part. 

QUESTIONS. 

(1) A body weighing 20 lbs. rests upon a horizontal surface upon 
which its coefficient of friction is 0.2. How great is the force re- 
quired to keep the body moving along the surface ? 

(2) It requires a force of 20 lbs. to keep a certain body moving 
along a horizontal plane, the coefficient of friction being 0.3. What 
is the weight of the body ? 

(3) A sledge weighing 10 lbs. can be drawn along a certain level 
surface by a force of 0.25 lb. How great may we expect the force 
to be which will just maintain motion when a load of 50 lbs. is placed 
on the sledge ? 

(4) A sledge weighing 50 lbs., having runners 1 in. wide, is 
dragged along a floor by a force of 15 lbs. How great a force would 
be required if the runners were twice as wide ? 

(5) According toRankine's Civil Engineering, the coefficient of slid- 
ing friction of loose earth on earth may be as much as 1, although it 
is generally less. Suppose a bank of earth, with 1 for the coefficient 
of friction, to be made of such steepness that the outer surface, if 
started, will continue to slide downward. 

(a) If a pole reaches 10 ft. straight downward into such a bank, 
how far along a horizontal line is the lower end of the pole from the 
surface ? 

(b) How great is the angle which the surface of such a bank makes 
with a horizontal plane ? 



CHAPTEE VII. 

THE PENDULUM. 

85. Use in Clocks. — Before leaving the subject of 
Mechanics and going to that of Light it is well to learn 
something about pendulums, which are used to control the 
motion of clocks. 

If you were to examine the works of an old-fashioned 
clock, you would find the power which drives it in a heavy 
weight working upon a kind of pulley by means of a long 
cord, but the device which governs the speed of the works 
and allows the motion to be neither too fast nor too slow is 
the pendulum. As a crowd of men at a turnstile, however 
they may try to force their way, can pass no faster than 
the swinging turnstile permits, so the clock-weight, which 
if the control were removed would run down at once with a 
furious buzzing of the wheels, is allowed by the pendulum 
to descend only very slowly, a very little distance at every 
swing of the pendulum, and not at all when the pendulum 
does not move. 

The rate at which the clock-wheels can move, then, 
depends upon the length of time required for each swing 
of the pendulum. We will try a few simple experiments to 
find out something about the laws of pendulum motion. 

EXPERIMENTS. 

Description of Apparatus. — A convenient method of suspending a 
simple pendulum is shown in Fig. 57, where B is one end of a wooden 
bar, which is bevelled off ou the side from which the pendulum 

86 



THE PENDULUM. 



87 



hangs. G is a cork fastened to the top of the bar and having in it a 
slit made by a sharp knife, through which slit the silk thread, S, 



o 
3 



o 
2 

Fig. 56. 






I 



FI&.57. 



passes. If this part of the thread is waxed, the fastening thus ob- 
tained holds the pendulum securely, although it is very easy to in- 
crease or decrease the length of the pendulum at will. The length 
of the pendulum is to be measured from the under side of the bar to 
the centre of the ball. It is intended that the length of No. 2 and 
No. 5 in Fig. 56 shall be the same as that of No. 1, that the length 
of No. 3 shall be one-fourth that of No. 1, and the length of No. 4 
one-ninth that of No. 1. It is therefore convenient to make the 
length of No. 1 just 36 inches, which will require 9 inches for the 
length of No. 3, and 4 inches for that of No. 4. The suspended 
body is a bullet in the case of each pendulum except No. 5, where 
it is some lighter object — a marble, for example. 

The whole apparatus is called No. XXI. 

(1) How does the time required for a single swing depend upon the 
lengthy or width, of the swing ? 

Set No. 1 and No. 2 swinging at the same instant and with the 
same width, or length, of swing, and watch them both for a little 
while until it is plain that under these circumstances they keep to- 



88 PHYSICS. 

gether, No. J. taking just as long a time for one swing, or for any 
number of swings, as No. 2 does. 

Then draw the ball of No. 1 about one inch aside from its position 
of rest, and the ball of No. 2 about fifteen inches aside from its 
position of rest, and release both balls at the same instant. Watch 
the two for some little time, a quarter of a minute or longer, and see 
whether at the end of that time they begin each swing together, as 
they did at first. If they do not, observe which one has gained upon 
the other, and, after one or two repetitions of the experiment, write 
down an answer to the question which the experiment was intended 
to meet. This answer should state which swing, the long or the 
short, if either, takes the longer time, and whether the difference in 
time is large or small compared with the time of either swing. 

(2) How does the time required for a single swing depend upon the 
length of the pendulum from the support down to the centre of the 
ball? 

Let one person, holding a watch in his hand, draw ball No. 2 
several inches aside from its position of rest and, releasing it at a 
convenient moment, give a signal to the class, and let the class 
count the number of single swings till, at the end of 20 seconds from 
the start, a signal is given to stop counting. 

In a similar manner the number of swings of No. 3 in 20 seconds 
and the number of swings of No. 4 in an equal time are found, and 
the observations for the three pendulums are recorded in a table, as 
follows : 

Number Time of Length Square Root 

of Swings. One Swing, of Pend. of Length. 

36 6 

9 3 

4 2 

The numbers to fill the fourth column must be found from those 
in the second and third columns. A comparison of the fourth col- 
umn with the sixth column will probably show that there is a close 
relation between the time of swing and the length of a pendulum.* 

(3) Weight of Pendulum-ball. — Finally, a comparison of No. 1 and 
No. 5, set in motion at the same time and with the same w T idth of 

* It is interesting and even amusing to watch pendulums 1 and 3 
or 3 and 4 swinging at the same time, both being started at the end 
of a swing at the same instant. 



Pendu- 


Whole 


lum. 


Time. 


No. 2 


20 sec. 


" 3 


(( 


" 4 


(C 



THE PENDULUM. 89 

swing, will show whether the time of swing depends much upon the 
nature of the suspended body. 

It will doubtless be noticed that the width of swing of the lighter 
body diminishes more rapidly than that of the heavier one. This 
gradual loss of motion is due to the resistance of the air. The re- 
sistance is about the same for both bodies if they have the same size, 
shape, and velocity, but a light body is more quickly stopped by a 
given resistance than a heavier body. This is the reason why one 
cannot throw an acorn or a piece of cork so far as one can a stone of 
the same size. 

86. Springs in Place of Pendulums. — It has been said 
above that pendulums are used to control clocks, but many 
clocks and all watches are controlled by means of vibrating 
springs; for these, like pendulums, are very regular in 
their swings and so are good time-keepers. The controlling 
springs (see the " balance " of a watch, Second Part) must 
not be confused with the much larger driving-brings^ or 
" matw-springs, " which are used in watches and in most 
clocks of the present day. 

More will be said about pendulums in the Second Part. 

QUESTIONS. 

1. If one pendulum is 9 inches long and another is 64 inches long, 
how will the time of vibration of the first compare with that of the 
second ? 

2. If pendulum A, 39 in. long, vibrates once in a second and 
pendulum B vibrates once in 5 seconds, what is the length of B ? 



CHAPTEE VIII. 

NATURE OP LIGHT : VISIBILITY OF OBJECTS. 

87. Light is Something that Travels. — We say that a 
lamp gives, or gives out, light. This is true. Light is 
something that comes to our eyes from any object and 
enables us to see the object. 

A substance through which light can travel is called a 
medium for light. We have ways of measuring the time 
required by light to travel a given distance in air and in 
many other media. 

88. Measurement of the Velocity of Light. — One of the 

simplest methods for measuring the velocity of light is that 
devised by the French physicist Fizeau. 

It consists essentially of a source of light, from which a 
bright beam may be obtained, a toothed wheel which may 
be made to revolve in a plane at right angles to the course 
of the beam of light, and a plane mirror. Apparatus is 
provided by means of which the rate at which the wheel 
revolves can be exactly measured. 

The beam of light passes through the space between two 
adjacent teeth of the wheel, travels a distance of several 
kilometers, is then reflected by the mirror, and returned 
over the same path by which it passed out. If the wheel is 
at rest, the beam as it returns will repass the aperture 
between the teeth through which it passed out. But it is 
easy to see that if the wheel could be revolved fast enough 
a tooth might be brought into the path of the returning 

90 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 91 

rays in time to intercept them. Still more rapid revolu- 
tions would bring a new gap between teeth into the path 
of the returning rays, and so on. In fact alternate eclipses 
and appearances of the returning rays are produced when 
the wheel is revolved at a high and continually increasing 
velocity. From the rate of motion of the wheel and the 
distance traversed by the beam it is not difficult to calculate 
the velocity of light. 

As a result of measurements made by somewhat different 
means from those just described, the velocity of light has 
been ascertained to be about 300,000 kilometers, or 186,000 
miles, per second in a vacuum. The velocity in air is a 
little less. 

89. Light is of Various Kinds. — Light as it comes from 
the sun, or from most lamps, is of many different kinds, all 
blended together so that the eye does not distinguish one 
kind from another; but when this mixture of light falls 
upon certain objects, pieces of glass called prisms, for in- 
stance, the mixture is broken up and we see the different 
colors, 

EXPERIMENT. 

Hold a glass prism (No. XXXI) in the direct sunlight in such a 
position that light after passing through the prism will fall upon a 
white surface not in the direct sunlight. 

This breaking up of light is considered further in § 134. 

90. Light a Wave-motion. — Before the nineteenth cen- 
tury many people believed light to consist of particles of 
matter, actually shot out in some way from the luminous 
body. These supposed particles were called corpuscles 
(that is, little bodies), and this theory as to the nature of 
light was called the corpuscular theory. 

We now believe that light is not a substance, but a kind 
of wave-motion, a shiver, which is sent along through 
bodies with great velocity and to very great distances, 



92 PHYSICS. 

although the particles of the body, or medium, transmitting 
this wave-motion travel very small distances on either side 
of their positions of rest. More will be said about this in 
the Second Part of this book. 

91. Color and Wave-length. — The different kinds of 
light, which produce in us the sensations of different colors, 
are distinguished from each other by differences of wave- 
length. Waves which produce the sensation of red, and 
which we often call red waves, are longer than the so-called 
blue waves, which produce the sensation of blue. One tint 
of red has a wave-length of one thirty-thousandth part of 
an inch. One tint of blue has a wave-length of one fifty- 
five-thousandth of an inch. 

92. Light Travels in Straight Lines.* — When direct 
sunlight enters a darkened room through a small hole, one 
can usually trace its course and boundary in the room by 
means of the air-borne dust particles which are lighted up 
by it. It is easy to see that the boundary, the side, of the 
learn of light is straight. This is one of the familiar facts 
which show that light travels in straight lines. Practical 
applications of this property of light are found in the. prac- 
tice of sighting rifles, cannon, and other firearms; in the 
method of glancing along the edge of a board, which the 
carpenter adopts to see whether it is straight; and in the 
various surveying operations, in which points are located by 
sighting with the unassisted eye, or by means of fine slits 
in metal plates, or by the aid of small telescopes. 

93. Light " Pencils and Rays." — If a beam of light is 

* This statement holds good only in cases in which the light 
travels in a medium or substance of uniform composition through- 
out. Even under such circumstances there are certain exceptions to 
the general rule of rectilinear propagation. These occur where 
light passes close by the edges of objects, but the effects produced, 
although very interesting and beautiful, are not sufficiently promi- 
nent to make their study in this book necessary or desirable. 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 93 



slender, it is a pencil of light. If the pencil is very slender 
indeed, it is called a ray of light, and is represented in 
drawings by a single line. 

94. Camera Obscura^ — This name means dark chamber. 

EXPERIMENT. 

Push the small tube of No. XXIV, closed end foremost, into the 
larger, and then, pointing the apparatus toward a window, look into 
the smaller tube and move it back and forth in the other till the best 
image of the window or of objects outside is obtained. 

It is evident at once that the image is upside down, that 
is, that the bottom of the image represents the top of the 
object. This is due to the fact that the light-rays, coming 
from the object and traversing the very small aperture in 
the end of the tube, cross each other in their passage, as in 
Fig. 58, where the object is represented by the arrow AB. 




Fig. 58. 

For instance, the cone of rays A A! from the tip of the arrow 
and the cone of rays BB' from the other end, cross at mn, 
and appear in the image at the spots A' and B' respectively. 
If the aperture mn were gradually made larger, the spots 
A! and B\ illuminated from A and B respectively, would 
grow larger and larger. The same would be true of the 
spots illuminated from other points of the arrow; and at 
last the growing spots would so overlap each other that the 
image would be lost in a mere blur of light on the screen. 



94 



PHYSICS. 



95. Shadows. — From the fact that light travels in 
straight lines, it is easy to see that it will be cat off from a 
portion of space behind any illuminated opaque object, just 
as waves of water are cut off by a breakwater, leaving a 
region of calm water behind it. The simplest case is that 
in which the light-giving object is as small as possible. 

EXPERIMENT. 

Light a bat- wing gas-jet or a kerosene lamp with a broad, thin 
flame, and cast the shadow of a lead-pencil, held vertical, on a sheet 
of white paper, having first the edge and then the broad side of the 
flame toward the pencil. Note the great difference in the sharp- 
ness of outline in the two cases. 

96. Umbra. — A shadow with a perfectly sharp outline 
could only be obtained by using as the source of light a 
mere point. To illustrate what would be the result if this 




Fig. 59. 

could be done, the student should examine Fig. 59. 

Of the light-rays proceeding from the point £, the cir- 
cular opaque object OF intercepts all which strike its sur- 
face, thus forming a shadow whose shape is in this case the 
frustum of a cone, OS TV.* The black space *STin the 
screen is not the whole shadow, but a section of the entire 
shadow OS TV. A perfect shadow like this, equally dark 
at all points, is called an umbra. 

97. Penumbra. — Suppose now that the source of light 
is of appreciable size, a candle-flame, for example: then the 

* That is, a cone with its top sliced off by a section parallel to the 
base. 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 95 

opaque object eats off all illumination from some por- 
tions of the screen, and from other portions cuts off only a 
part of the light, as in Fig. 60. 

That part of the screen which receives light from part of 
the flame AB, but not from all of it, will appear a partially 
shaded ring, P'S'SP, around the central area of total 
shadow. This ring forms what is known as the penumbra 
(from two Latin words meaning almost and shadow). 

On account of the comparatively large size of most 
sources of light most shadows are surrounded by a wide 
margin of penumbra. The student will find the best 
examples of clear-cut shadows in those cast upon near sur- 
faces by opaque bodies exposed to electric arc-lights, and he 




Fig 60. 

may compare the dim and indistinct shadows of the leaves 
of shade-trees exposed to the sun, with those cast by the 
same objects exposed to the electric light at night, in which 
even the serrated margins of the leaves are sometimes clearly 
outlined. 



98. How Light Weakens with Distance . Law of Inverse 
Square- — If two equally large surfaces are turned toward 
a very small flame, distant 1 ft. from one and 2 ft. from 
the other, the nearer surface will receive very nearly four 
times as much light from the flame as the more distant but- 



96 PBTS1CS. 

face. It is easy to prove that this is true if light travels 
in straight lines diverging from a point (Second Part). 

If one surface is three times as far away as the other it 
will receive only one ninth as much light as the nearer 
one, and so on. 

The law, which holds when the diameter of the light- 
giving spot is very small compared with the distance from 
it to the receiving surface, may be stated thus : The amount 
of light received on a surface of given area from a given 
source of light is inversely proportional to the square of the 
distance from the source to the surface. 

This is called the law of inverse square. 

It follows from this law that if a lamp L sends ,to a given 
surface at a distance D 311st as much light as another lamp 
U sends to the same surface at a distance D\ the light- 
giving powers of the two lamps, which powers we will call 
P and P\ must be such that 

P: P':: D 2 : D'\ 

Illustration. 
A candle-flame 30 cm. from a white card and an incandescent elec- 
tric lamp 120 cm. from the same card light it up equally. What is 
the relative power of the two sources ? 

Pi (for the lamp) : P c (for the candle) : : 120 2 . 30 2 . 
Hence P = P c X 16. 

99. Photometry ; Rumford's Photometer. — It is a matter 
of great practical importance to compare the illuminating 
power of different lamps. This operation is called photom- 
etry, or light-measurement. It cannot be done by merely 
observing the lamps directly ; for the eye is unable in this 
way to distinguish slight differences of power, and if the 
lights are of somewhat different colors the unaided eye gives 
only the vaguest indications in regard to their comparative 
efficiency. 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 91 

One of the simplest devices for measuring the relative 
power of two sources of light is RuniforcTs photometer, 
which compares the shadows cast by a rod placed in front 
of them. 

EXERCISE 16. 
USE OF RUMFORD PHOTOMETER. 
Apparatus : Two small kerosene lamps like No. 33. A cardboard 
screen and its support (No. 32 and No. 21). Any opaque rod about 
1 cm. in diameter and 10 or 15 cm. tall, supported upright ; e.g., No. 
13 standing in a hole bored m a small block, or a Bunsen burner, A 
meter rod. 

The ooject of the experiments will be to find whether a flame sends 
more or less light from its broad side than from its edge, and, if so, 
how much, The flame should be made as large as they can well be 
without smoking. 

The apparatus should be arranged as in Fig. 61, 

A 



C 



Fig. 61. 



Jl 



L is one of the lamps to be compared, and L the other ; R the 
rod AB the screen, and Sl and 8& the shadows. The lamps should 
be so arranged that lines drawn from their centres to the centre of H 
will make nearly equal angles at i? with the line CD, drawn at right 
angles to the screen through the centre of i?, and on this line the 
observer should be placed. The shadows should be near each other, 
but must not overlap. 

It is plain that the shadow corresponding to L is illuminated by 
light from L' and that the one corresponding to U is illuminated 
by light from L. 

Place the lamps equidistant from the rod. and, shielding the eyes 



98 PHYSICS. 

from the direct light of the flames, adjust the flames, turned edge- 
wise to the rod, until the shadows are of equal darkness. 

Then turn one of the lamps about in place until its flame is flat- 
wise to the rod, and compare the shadows again, fixing the attention 
upon the middle of the more blurred one. If the shadows still appear 
to be of equal darkness, record the fact. If they do not, move one of 
the lamps toward or from the rod until the shadows appear equally 
dark, and then record the distance of each flame from the corre- 
sponding shadow. 

Try each lamp in turn flatwise, the other being edgewise. Be- 
tween the trials test the flames again in their original position, to 
make sure that they are still equal. 

If, on the whole, it appears that one aspect of the flame, broad side 
or edge, is more effective than the other, estimate the relative light- 
giving power of the two aspects from the measured distances, mak- 
ing use of the law of inverse square. 

100. Bunsen's Photometer. — The form of photometer 
devised by the German chemist and physicist Bunsen 
yields, under suitable conditions, more accurate results than 
the apparatus just described, and is equally simple, but 
more difficult to use in an undarkened room. 

EXPERIMENT. 

Drop a little paraffin on a sheet of heavy, unsized, white paper, — 
thin drawing paper, for instance. Heat the paper by placing on it a 
moderately hot iron weight or a can of hot water, until the paraffin 
is entirely melted and soaked evenly into the paper, so as to make a 
roughly circular spot about 3 cm. in diameter. Cut out of the paper 
a circle about 12 cm. in diameter with the spot just prepared in its 
centre. It will be noticed that the spot is translucent ; that is, it 
allows some light to pass through it, although objects cannot be 
clearly seen through it. If one looks from a darker portion of the 
room toward a brigther portion with this screen interposed, the 
translucent spot will appear brighter than the ring of opaque paper 
around it, while, under the reverse conditions of illumination, the 
opaque ring will appear brighter than the spot. 

Mount the screen in any convenient way ; for example, in a block 
(No. 21). In making photometric observations with this screen the 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 99 

illumination from one source of light is to be allowed to fall at right 
angles on one side of the screen, and that from the other source is to 
fall at right angles on the other side. The screen is then to be 
moved back and forth between the two lights until a position is 
found in which the appearance of the screen, as tested by the con- 
trast between the central spot and the rest of the surface, is exactly 
the same on both sides when viewed from the same angle. The 
illumination on the two sides is then equal, and the distances from 
the lights to the screen will afford a means of comparing the power 
of the lights, as already indicated in § 98. 

It is hardly worth while to attempt this experiment in an 
undarkened room. 

101. Effect on Light of the Body on which it Falls. — 

When light-rays meet the surface of a body they may be : 

a. Regularly reflected : that is, sent off from the surface 
in a direction which can be calculated or foretold, if we 
know the direction in which they are to strike the surface, 
as sunlight is reflected by a mirror. 

b. Irregularly reflected or scattered: that is, sent back 
or off from the surface in many different directions, as sun- 
light is sent back from the surface of white cloth or white 
paper. 

c. Transmitted : that is, allowed to pass through as sun- 
light through clear window-glass. 

d. Absorbed : that is, neither reflected nor transmitted, 
but swallowed up, as sunlight by a lamp-black surface 
upon which it falls. 

It usually happens that more than one of these effects is 
produced by the same body at the same time. 

102. Visibility of Objects.* — Very few of the objects we 
see shine by their own light, as we can tell by testing them 
in the dark. They merely give off the light, or some part 

* For much interesting and valuable matter upon this subject see 
Rood's Text-book of Colo?', Appleton & Co. 



100 PHYSICS. 

of the light, which has fallen upon them from the sun, or 
from some other light-giving body. 

Of course we see many things every day upon which 
neither the sun nor any lamp is directly shining. We see 
them by what is called "daylight. 53 This, however, is 
sunlight, although it may not have come straight from the 
sun to the objects that we see lighted up by it. It may 
have gone from the sun to a mass of clouds, from the clouds 
to the surface of fields or streets or walls of houses, and 
from these surfaces into corners where the sun itself is 
never seen. 

It is extremely fortunate for us that all external objects 
do not treat the light which falls upon them in exactly the 
same way. If they did, all things would be of one color, 
and we could distinguish only light and shade. We have 
something like this condition after a fresh fall of snow 
which has covered roofs and trees as well as the ground. 
There are always, however, parts of trees and houses not 
completely covered by the snow, and this fact enables us to 
keep our bearings fairly well. If everything were covered 
by the snow, our eyes would not be of much more help to 
us in broad daylight than they are in the dead of night. 

103. Colors of Transparent Bodies. — Colored pieces of 
glass, colored liquids, and other transparent bodies, gen- 
erally owe their color to the fact that they are not trans- 
parent to all kinds of light. The light which enters them, 
sunlight, for example, usually consists of many different 
colors blended together; and they rob this light of those 
colors which suit their own constitution, transmitting the 
rest. It is the transmitted, the rejected, light which we get 
from them that gives them their apparent color. The 
light which they absorb is turned to something else in the 
absorption, and is no longer light. It is usually turned 
into heat. 



NATURE OF LIGHT: VISIBILITY OF OBJECTS. 101 

104. Colors of Opaque Bodies. — Most bodies with which 
we art? familiar do not appear to transmit light. We cannot 
see through them, and we call them opaque bodies. 

In fact, most so-called opaque bodies are not perfectly so. 
If they are made into very thin sheets, the sun can shine 
faintly through them. Even when they are in thick 
masses, the light penetrates a very little distance beneath 
the surface, where some of it is absorbed, and some, being 
reflected by interior particles, returns to the outside. This 
returning light is usually different in color from the mix- 
ture of lights that entered, certain parts having been 
absorbed more than others. 

105. Light from Surface of Colored Bodies. — The light 
reflected from the real external surface of non-metallic 
colored bodies receiving white light is usually not colored. 
The following experiment shows an illustration of this fact: 

EXPERIMENT. 

Let a beam of direct sunlight, entering a window, fall very ob- 
liquely upon a sheet of colored glass in such a way that the reflected 
beam will fall upon a white surface. Observe the color of the 
reflected light. 

Certain materials, silks, for example, may reflect white 
light, from the outer surface, together with considerable 
colored light that has penetrated this surface and has been 
sent back from the interior. The white light gives the 
sheen, but in the spots where this is strong the color is not 
at the same time very evident, being made to look pale by 
the large amount of white light mixed with it. 

The following experiment will show how the color coming 
from an object maybe deepened by diminishing the amount 
of white light reflected from the external surfaces of its 
numerous particles. 

EXPERIMENT. 

Grind a lump of sulphate of copper to a fine powder and observe 
how faint the blue color becomes ; then wet the powder with water, 



102 PHYSICS. 

which adds nothing but prevents some of the external reflection, 
and note the decided deepening of the blue. 

In velvet * the ends of the fibres, which reflect but little 
white light, are turned outward, and the light which pene- 
trates the surface and then returns to the outside is deeply- 
colored. 

* See Rood, p. 79. 



CHAPTEE IX. 

REGULAR REFLECTION OF LIGHT. 

106. Reflectors. — Smooth, even surfaces, like the surface 
of still water, polished glass, or polished metal, reflect light 
regularly (§ 101). 

Transparent reflectors are not convenient for ordinary 
use: partly because light which we do not want may come 
through them from behind; partly because they reflect 
really well only such light as falls upon them very obliquely. 

EXPERIMENT. 

Let M t Fig. 62. be a piece of clear window-glass, L a lamp, and E 
the position of the observer's eye. The rays LM and J/^rnake a 

W *T 




large angle with the line 3IX, which is the normal to the surface of 
the glass. 

Observe the comparative brightness of the flame itself, and its 
picture or image, seen by reflection from 31. Xotice with what 
degree of clearness objects back of 31, as, for instance, points on the 
wall WW, can be seen through 31 in the direction E3I. 

Gradually move the lamp and the eye toward the point X, until at 
last both lamp and eye are as nearly as possible on the line X3L 
While making these changes of position observe any resulting 
changes in the brilliancy of the image of Z, and in the clearness 
with which objects on the line WW are seen through the glass. 

103 



104 PHYSICS. 

The reflecting surface which we make use of in a common 
mirror is not the front surface of the glass, but the metallic 
surface at the back. The glass is merely a convenient 
transparent support for the metallic layer, keeping it in 
shape and protecting it from being tarnished, as it soon 
would be if exposed to the air. 

Reflection from a Plane Mirror. 
107. Where the Image Is. — A plane mirror is a flat 
mirror. We shall study curved mirrors later. 

When we place an object in front of a plane mirror and 
stand in a proper position we see an image, or "reflection," 

.^____ °^ the object, and we say that we 

see the object, or its image, in 

~ the mirror. If M, Fig. 63, is 

•P t »P A the mirror, a point of the 

•P 2 .p 3 object, and P 1? P 2 , P 3 , and P 4 

are the positions of four eyes, all 

fig. 63. may see at the same time an 

image of the point in the mirror. Our first Exercise in 

light is intended to answer the question whether all these 

eyes see the same image, that is, whether all are looking 

toward the same point, and if so, where this point is — in 

front of the mirror, or behind it, or at its surface. 

EXERCISE 1 7. 

IMAGES IN A PLANE MIRROR. 
Apparatus : A mirror (No. 23). A rectangular block (No. 9). A 
rubber band to hold the mirror to the block. Two straight- edged 
wooden rulers (Nos. 24a and 24b). A meas- 
uring-stick (No. 3). A sheet of thin white 
paper about 12 inches by 20 inches. A small 
block (No. 25). Attach the mirror to the 
large block by means of the rubber band in 
the manner shown by Fig. 64. 

-^ .i ., n Fig. 64. 

Draw a straight pencil -mark across the 

sheet of paper at its middle, and set the back surface of the mirror 




BEGULAR REFLECTION OF LIGHT. 



105 



\7* 

3 



directly over and parallel to this line, the middle of the mirror being 
very near the centre of the sheet. See Fig. 65. 

Draw on the sheet of paper in front of the mirror a triangle, each 
side of which shall be several inches long, and no corner of which 
shall be less than three inches from the mirror. 
It is well to have one angle of the triangle not 
directly in front of the mirror, but somewhat to 
one side, like point No. 1 in the figure. 

Place the small block in such a position that 
the vertical pencil-mark which it bears shall be 
directly over point No. 1 of the triangle. Then 
lay a straight- edged ruler (Fig. 66), upon the 
paper in such a position that one of its long 
horizontal edges, PQ, shall point directly to- 
ward the image of the vertical pencil-mark, as' 
seen in the mirror.* The ruler should be so 
placed that the line of sight will strike near 
one end of the face of the mirror. Then with 
a well-sharpened pencil draw upon the paper a 
fine clear mark alongside that edge of the ruler 
which lies just beneath the line PQ (Fig. 66) along which the 
sight has been taken. Mark this line 1, because it points toward the 
image of the vertical pencil-mark when this 
mark is over point No. 1. 

Next, without disturbing anything else, 
Fig. 66. p i ace the ruler in a new position, far removed 

sidewise from the position just occupied, sight as before, draw 
another line alongside the ruler, and mark this line also 1. 

Then with the ruler in a new position, about half-way between 
the first two, if this is convenient, draw a third line in the same 
way, and mark this also 1. 

All this time the small block has remained unmoved, and the 
pencil-mark upon it has pointed straight down at point No. 1. 

* Many persons cannot do this at first unless they are especially 
instructed. A person who is not near-sighted should hold his eye 
eight or ten inches distant from P, and should then direct the ruler 
in such a way that the point P, the point Q, and the image of the 
vertical pencil-mark seen in the mirror shall all lie in one straight 
line. Do not try to look along the vertical side of the ruler, but hold 
the eye high enough to see all the time the top of the ruler 



Fig. 65. 



106 PHYSICS. 

Now place the small block so that the pencil mark shall point 
straight down at point No. 2. While it is in this position draw 
three straight lines toward the image and mark each one of these 2. 

Finally, put the pencil mark over point No. 3, draw three straight 
lines toward its image, and mark each of them 3.* 

When the three sets of lines Lave been drawn, the two blocks and 
the mirror are removed from the paper, and each line is then length- 
ened f until it cros es both the others of the same set ; that is, each 
No. 1 line is continued toward or beyond the mirror till it crosses 
the two other No. 1 lines. Then the No. 2 set and the No. 3 set are 
treated in the same way. 

After each set of lines has been extended in this way, it will be in 
order to answer the qu* stion whether all the lines of any one set 
lead to the "same point or nearly so, and, if so, where is this point 
situated with respect to the mirror and to the point whose image it is. 

If the image of each point, No. 1, No. 2, and No. 3, can be thus 
found, connect the image-points with each other by straight lines^ 
and thus make an image-triangle. 

Then fold the sheet of paper carefully along the pencil-mark by 
which the mirror was placed, and holding the folded sheet against 
a window, so that the light from without will shine through it, 
compare the size and shape of the two triangles and their relative 
positions with respect to the line along which the paper is folded. 

* While drawing all these lines the experimenter should look fre- 
quently to see whether the back of the mirror remains in place. It 
may be thrown out of place by a little blow or by rubbing the paper 
hard to remove pencil-marks. 

f If a line has to be extended far it is well to use two rulers, A 
and B, as shown in Fig. 67. First A is put into position and a line 



~7T 



Fig. 67. 

is drawn alongside it. Then, while A remains unmoved, B is care- 
fully brought close to it, as the figure shows ; then B is held firmly 
in place whil^ A is pushed forward to tLe position indicated by the 
dotted lints. B is then removed without disturbing A, and again a 
line is drawn alongside A. In this way a line may be continued 
nearly straight for a considerable distance, 



REGULAR REFLECTION OF LIGHT. 



107 



'I he general rule for placing the image of any point should be 
recorded when it is found. 

The final result aimed at in this A B 

Exercise should be to enable the 
student to tell, without farther ex- 
periment, in any new case given 
him (Fig. 68, for instance, in which 
A B is the line upon which the 
mirror stands), the position of the 
image of points No. 1, No. 2, No. 3, 
and No. 4, and so the shape and 
position of the image of the figure 
at the corners of which these points 
lie. 

108. The Law of Reflection.— In Fig. 69, MM is a 
I\ 





mirroi surface, CD a normal to this surface, OC a ray in- 
cident at the point C, and CP the same ray after reflection. 

The angle i is called the angle of incidence. 

The angle r is called the angle of reflection. 

The "law of reflection" is, that the angle of reflection 
is equal to the angle of incidence. 

This law is easily proved on the basis of what we have 



108 PHYSICS. 

learned in the preceding exercise. The line of proof is this : 
The image of is at I. The angles at E are right angles ; 
EI — EO ; EC is common to the two triangles ; hence the 
triangle CEI is similar to the triangle CEO. Then angle 
% = angle EOC = angle EIC = angle r. 

109. Real and Unreal Images. — If the rays of light pro- 
ceeding from a point are by any means really brought 
together again at a different point, as in Fig. 79, then the 
second point is called a real image of the first. A real 
image has an actual existence in space, and will show as a 
picture upon a properly placed white screen. 

If the rays of light proceeding from a point are by any 
means made falsely to appear to diverge from a different 
point, as in Fig. 70, then the second point is called an 



/ / v V 




Fig. 70. 

unreal, or virtual, image of the first. A virtual image has 
no real existence in space, and would not show upon a screen 
placed where it appears to be. 

Evidently the image formed by a plane mirror is an un- 
real image. 

110. Images of Images. — If any of the rays from (Fig. 
71) after reflection from the mirror A fall upon a second 
plane mirror i?, they will be treated by this second mirror 



REGULAR REFLECTION OF LIGHT. 



109 



just as if they really came from I x \ that is, we shall, look- 
ing into the mirror B in the right direction, see an image 
of the image /„ and this second image, 7 2 , will appear just 




Fig. 71. 

as if it were the image of an actual object, sending rays 
from I r 

The rays reflected first from A and next from B might 
then fall upon a third mirror, and give an image of the 
image 7 2 , and so on; but at each reflection there is some 
loss of light, and an image formed 
after many reflections might be 
dim. 

111. Positions of the Various 
Images. — Let A and B in Fig. 
72 represent the positions of two 
plane mirrors meeting at right 
angles with each other at the 
point C. Let be a small object 
placed between the mirror faces. 

We shall have one image, 7 l3 formed by mirror A with- 



r 3*£_ 



Ji 



Fig. 72. 



110 PHYSICS. 

out any help from mirror L\ and another, 7 2 , formed by 
B without help from A. There is also 7 3 , the image of / l5 
seen in B\ and there is 7 4 , the image of 7 2 , seen in A, I b 
and J 4 fail at the same spot. 

We cannot with this arrangement of the mirrors get 
images of I 2 and / 4 ; because rays leaving mirror A as if 
diverging from I A would not strike the face of i?, and rays 
leaving mirror B as if diverging from 7 3 would not strike 
the face of A. 

Observe that and its images fill the corners of a rec- 
tangle. If were midway becween the mirrors, the 
rectangle would be a square, with at its centre. 

If the angle between the mirrors were made a bit less 
than 90°., I z and 7 4 would fall apart. If the angle were 
made 60°, one sixth of a circle, lying half-way between 
them, and its images would fill the corners of a regular 
hexagon having Cat the centre, 

If the angle were 30°, one tivelfth of a circle, and its 
images would fill the corners of a tweive-sided figure. 

EXPERIMENT. 

Place the hinged mirrors of No. XXV upon the board, with the 
reflecting surfaces making an angle of 90°, 
the point 1, Fig. 73, being midway be- 
tween them. Place a lighted candle, of 
such length that its flame will not be 
above the upper edge of the mirrors, ex- 
actly over the spot 1. 

Note the positions of the images of the 
candle seen in the mirrors. Put pegs into 
the holes behind the mirrors in such posi- 
tions that to an observer placed in front 
of the mirrors, so as to see the images in 
the mirrors and the pegs over the mirrors, the pegs will appear to 
coincide with the images. 

Then make the angle between the mirrors 60° and place pegs to 
coincide with the images. 
Finally, try an angle of 30°. 




REGULAR REFLECTION OF LIGHT, 111 

112. The Kaleidoscope. — The preceding passages give 
an explanation of the kaleidoscope, No. XXVI, with which 
most oeautiful effects of endless variety can be obtained. 
The kaleidoscope uses bits of colored glass instead of a 
candle flame, and sometimes has three mirrors put together 
at angles of 60°. 

QUESTIONS AND PROBLEMS. 

(1) In a lighted room at night the glass of a window will serve as a 
mirror. In daylight unsilvered glass with a black cloth behind it 
may be used in the same way. Can you explain this ? 

(2) Soon after the moons of Mars were discovered in 1877 some one 
announced in a newspaper that one of these moons could be seen near 
Mars by looking at the reflection of that planet in a common mirror. 
It is true that a faint bright speck appeared near the image of Mars as 
thus seen, which did not appear when the planet was looked at di- 
rectly, but the true moons could be seen only by the aid of powerful 
telescopes. Can you, after trying the experiment with any bright 
star, explain the appearance seen in the mirror ? 

(3) Write some short word as it would appear in a mirror if the 
printed page containing it were reflected in the mirror. 

(4) A person standing in the middle of a room 20 ft. wide looks 
with one eye into a mirror 2 ft. square set in the wall of one side of 
the room. How many square feet of the wall behind himself could he 
see reflected in the mirror if his own image did not obstruct the view? 

{Suggestion : Draw a diagram representing the position of the ob- 
server, the mirror, the reflected wall and its image, all on a horizon- 
tal plane.) 

(5) A candle-flame is placed half-way between two plane mirrors 
which meet at an angle of 40°. How many images appear, and how 
are they arranged ? 

Reflection from Curved Mirrors. 

113. Spherical and Cylindrical. — Most curved mirrors 
are parts of spherical surfaces. We shall, however, study 
mirrors which are parts of cylinders. They are more con- 
venient for our use than spherical mirrors, and they are less 
expensive. 




112 PHYSICS. 

We shall use both the convex, or bulging, and the concave, 
or hollowed, face of the mirror. 

114. Centre of Curvature, etc. — MM, Fig. 74, represents 
a cut through a cylindrical mirror at 
right angles with the straight lines of its 
^3i surface. This cut is of course a part of 
a circle. 

C, the centre of the circle, is called the 
centre of curvature of the mirror. 

The point is called the centre of the 
mirror. 

The line CO, extended to any distance 
fig. 74. in either direction, is called the principal 

axis of the mirror. 

Any straight line extending, like CR, through C and 
across the line MM, but not through the point 0, is called 
a secondary axis of the mirror. 

EXERCISE 18. 

IMAGES FORMED BY A CONVEX CYLINDRICAL MIRROR. 

Apparatus : The mirror (No. 27). A measuring-stick (No 3). Small 
block (No. 25). Rulers (No. 24 A and b). Sheet of white paper. 
The plane mirror (No. 23) and its supporting block (No. 9), 

Hold the mirror with its straight edges vertical, and look at the 
image of your own face in the convex surface. You will see that the 
image is distorted, appearing too narrow for its length. Hold the 
mirror with its straight edges horizontal, and the image will be dis 
torted in the opposite way, appearing too wide for its iength. The 
object of the following experiments is to give a better understanding 
of these curious effects. 

Set the mirror on the table and bring one end of the plane mirror 
close to the surface of the curved mirror, as in Fig 75 Then place 
the small block in front of both mirrors, as in Fig. 75, in such a 
position that you can see the block reflected in both mirrors at the 
same time. 

Do the two images thus seen appear of the same height ? 

Do they appear of the same width ? 



REGULAR REFLECTION OF LIGHT. 



113 



Fill out, if you can, the following statement : Lines of the object 
which are parallel to the straight lines of the cylindrical mirror 

appear in the cylindrical mirror 

plane mirror. 

Lines of the object which are at right angles icith the straight lines of 

the cylindrical mirror appear in the cylindrical 

mirror in the plane mirror. 




Fig. 75. 

Remove the plane mirror. Holding the base of the curved mirror 
firmly in place, make a fine, clear, pencil-mark on the paper along 
the outer edge of the mirror. Then mark on the paper the point (7, 
which is the centre of curvature of the mirror. 

About 5 cm from the front of the 
mirror draw an arrow 6 cm. long, 
marking the ends and the middle as 
in Fig. 76. Then place the small 
block so that the vertical pencil-mark 
which it carries will point straight 
down at point No. 1. 

With the straight-edged ruler draw 
two lines, well apart, toward the 
image of this vertical line as seen in 
the mirror, avoiding parts of the 
mirror, if there are such, that do not 
give a good image of the line. Mark 
each of these lines 1. Then draw two 
lines for point No. 2 and two for point No. 3, in the same way. 




2 > »>^> - 



Fig. 76. 



114 



PHYSICS. 



Then clear the paper and prolong each pair of lines till it conies to 
a crossing-point. The three points thus found will locate the images 
of object-points No. 1, Noi 2, and No. 3, respectively, and a line con- 
necting these three image-points will give an idea of the shape of the 
image-arrow, whether it is straight or not, and whether its curvature, 
if it has any, is in the same general direction as the curvature of the 
mirror or in the opposite direction. 

Draw a straight line from each marked object-point to the corre- 
sponding image-point, and prolong these three lines until they cross 
each other. Note where the crossing occurs. 

Is the image longer or shorter than the object ? Is it nearer to, 
or farther from, the mirror than the object is ? 

(It must be understood that the pupil is asked these questions only 
in regard to the particular case that he has tried. He cannot tell with- 
out further experiments or further instruction whether the answers he 
gives in this case would be true for all cases of objects reflected in 
mirrors such as he is using, for he does not know that the distance of 
the object from the mirror may not decide all these questions. The 
fact is, however, that, if he has found correct answers to the questions 
asked for his one case, the same answers will be true for the same 
questions in all cases with convex cylindrical mirrors. The effects 
seen with concave mirrors are much more complicated.) 




Fig. 77. 



115. "Law of Reflection " Still Holds.— With curved 
mirrors, as with plane mirrors, the law (§ 108) angle of in- 



REGULAR REFLECTION OF LIGHT. 115 

cidence = angle of reflection holds. With the help of this 
law we can see why the image of a point is nearer the 
mirror, when this is convex, than the point itself is. 

Let (Fig. 77) be the object-point in front of the convex 
mirror JO/, the centre of curvature being at C. A line 
drawn from C to any point of the mirror is at right angles 
with the mirror at the point of crossing. Two rays going 
from to the mirror-front appear after reflection to come 
from /, which is nearer the mirror than is. 

116. Principal Focus and Focal Length. — If rays come 
from some very distant point on the principal axis (§ 114), 
they are practically parallel to each other when they reach 
the mirror. Two such rays are represented by r 1 and r 2 
(Fig. 77). Applying the law of reflection to them, we find 
that after reflection they appear, as r/ and r 2 ', to diverge 
from a point P, which is very nearly midway between the 
reflecting surface and the centre of curvature- 

The point P is called the principal focus of the convex 
mirror. It may be defined as the point which marks the 
image of an object-point situated a long distance away from 
the mirror on the principal axis, or as the point from which 
rays coming to the mirror parallel to the principal axis 
appear to diverge after reflection. 

The distance, measured along the principal axis, from the 
principal focus to the reflecting face is called the focal 
length of the mirror. 

The principal focus and the focal length play a very im- 
portant part in the science of curved mirrors and lenses 
(§ 136). More will be said of this later. See § 124. 

117. Concave Mirrors. — If the concave side of the mirror 
were used, it is easy to see from Fig. 78 that rays from a 
point near the mirror-front would after reflection appear 
to come from a point /, which is farther from the mirror 



116 



PHYSICS. 



than is. It is evident that the rays from are more 
nearly parallel to each other after reflection than before. 

Rays from a point 0', somewhat farther from the mirror 
than 0, appear after reflection to come from a still more 
distant point, I\ and these rays are nearly parallel after 
reflection. It is easy to see that if the object-point were 
put somewhat farther still from the mirror, the rays pro- 
ceeding from it might, after reflection, be parallel to each 




other. They would appear to come from a point as far as 
possible behind the mirror. 

If the object-point is placed still farther away from the 
mirror, as at in Fig. 79, the rays may after reflection be 
actually converging, and cross at a point I in front of the 
mirror. This image I is a real image (§ 109), and if is 
bright enough, the image /may be seen, like a picture, on 
a piece of white paper or cloth placed in the right position. 



REGULAR REFLECTION OF LIGHT. 117 

We see that the centre of curvature lies between the 
object-point and the image-point / in the case shown by 
Fig. 79. This is always so in the case of real images 
formed by concave mirrors, unless the object-point is at 
(7, in which case the image-point also falls at C. 

If the object-point were placed where / now is, in Fig. 
79, the image-point would fall where now is. 

EXERCISE 19. 

IMAGES FORMED BY A CONCAVE CYLINDRICAL MIRROR. 
Apparatus: The same as in the preceding Exercise, and in addition 
a common pin. 

Preliminary . 

Remove the mirror from the base- 
board ; place the latter upon the paper* 
and mark on the paper the point 
and the curved outline of the board. 

Make the distance OA 4.2 cm., and 
draw the arrow A 4 cm. long. Draw 
radii from G through the ends of A. 

Make the distance GB 3.5 cm., and 
draw B from radius to radius. 

Make the distance GD 1.5 cm., and 
draw D from radius to radius. 

(All this should be done before the 
regular Exercise begins.) 

Place the mirror in position as in 
Fig. 80, and, keeping the eye about 20 
cm. from it, look at the images of A, B, and D. 

Do the images of A and B point in the same general direction, from 
left to right in the figure, as the arrows themselves ? 

Is the same answer true of D and its image ? 

Are the images of A and B longer or shorter than the arrows them- 
selves ? 

At the centre of A stand the pin upright, and laying the two rulers 
on the paper, point one edge of each toward the image of the pin, 
contriving to have these edges make a considerable angle with each 
other. In this way the position of the image is located. Is it behind 
the mirror or in front ? Is it, then, a real image or an unreal one ? 




118 



PHYSICS. 



By the same method locate the image of the pin when erected at 
the centre of B and when at the centre of D, asking and answering 
in each case the same questions that were asked when the pin was at 
the centre of A. 

If time permits continue the Exercise as follows : 

Extend the two radii r r by the lines r' r' drawn on the paper, as in 
Fig. 80. Draw the arrow E, 6 or 8 cm. distant from G, marking 
points 1, 2, and 3, upon it. Locate the image of each of these points 
by the method used in the preceding Exercise with the convex side 
of the mirror, drawing upon the paper the lines of sight and the 
image of the arrow. 

118. Principal Focus of Concave Mirror. — Tlie principal 
focus of a concave mirror is the point to ivhich rays, coming 
to the mirror parallel to the principal axis, converge after 
reflection. In other words, it is the point which marks the 
image (real) of a very distant point on the principal axis. 

As in the case of a convex mirror, the principal focus lies 
very nearly midway between the reflecting surface and the 
centre of curvature. 

119. Rule for Placing Images. — Fig. 81 illustrates an 





Fig. 81. Fig. 82. 

easy rule for finding the position of an image in a convex 
mirror. 

Let AB be the object. Draw one ray from A straight 
toward the centre of curvature. This ray will return on 



REGULAR REFLECTION OF LIGHT. 119 

itself after reflection, coming as if from C. Draw another 
ray from A parallel to the principal axis. This will after 
reflection appear to come from P, the principal focus 
(§ 116). Both reflected rays appear to come from A\ 
which is therefore the image of A. 

The image of B is found in the same way. 

If the object is a straight line, as in this figure it is cus- 
tomary to represent the image by drawing a straight line 
from A' to B '. This is inaccurate, as Exercise 18 should 
show. 

Fig. 82 shows the same method applied to a concave 
mirror. 

120. Distorted Images. — In Exercises 18 and 19, and in 

all the figures that have been given representing cylindrical 
mirrors, we have been dealing with rays which are, both 
before and after reflection, parallel to the plane on which 
the mirror rests. If we make use of other rays, as we do 
when looking obliquely down at the mirror face, we see 
things sadly twisted, the effects thus obtained being too 
difficult for oar profitable study. 

121. Relation of Cylindrical to Spherical Mirrors. — If 

we were to use a spherical mirror, placed with its principal 
axis horizontal, and employ only horizontal rays striking 
the mirror on a narrow horizontal strip through its middle, 
we should get effects quite like those we have already 
studied. All the figures from 71 to 82 would apply as well 
to a spherical mirror so used as to a cylindrical mirror. 
Indeed, these figures are like those commonly given to show 
the effects obtained with spherical mirrors. 

For general use spherical mirrors are better than cylindri- 
cal mirrors, because they can be used from more points of 
view without giving badly distorted images. 



120 PHYSICS. 



EXPERIMENTS. 



With a concave spherical mirror 5 or 6 inches wide (No. XXVII) 
3f-~~" M interesting lecture-table experiments may be 

made in a slightly darkened room, the image 
of a candle-name or, better, gas-flame being 
thrown upon a screen so as to be visible to all 
in the room. The screen should be uf tracing 
Lq cloth or oiled paper, so that the image upon it 

S may be seen from both sides. An opaque screen 

q 9 should hide the flame itself from the eyes of 

the class. Fig. 83 suggests a good arrange- 
ment, MM beng the mirror, G its centre of cur- 
vature, L the flame, S the opaque screen, and 
S' the tracing- cloth screen. 

The positions of L and S may be greatly 
varied and may be interchanged, but the least 

g' distance of either from mirror the should be 

Fig. 83. rather more than one half the radius of curva- 

ture of the mirror, if real images are desired. 

122. Principal Use of Spherical Mirrors. — Although 
spherical mirrors are sometimes used to form images, as in 
certain telescopes, probably their most important use is to 
concentrate light upon some object that cannot otherwise 
be well seen. 

Thus, the small objects which are to be looked at with a 
microscope need to be brightly illuminated, and a concave 
mirror is commonly used to throw light upon them. 

123. The Ophthalmoscope. — Often a physician wishes to 
see what is wrong in the depths 
of a patient's eye. To do this 
tfie interior of the eye must be 
especially lighted up. If this 
is done by holding a flame in 
front of it, the flame dazzles the 
eye of the observer and therefore 

is of little use. The difficulty is overcome by means of the 




REGULAE REFLECTION OF LIGHT. 121 

ophthalmoscope, Fig. 84, where M is a curved mirror with 
a hole in the centre; L is some source of light, placed so 
that the rays proceeding from it to the mirror pass by re- 
flection into the eye of the patient, represented by E\ and 
marks the position of the observer's eye. 

This simple application of the concave mirror was made 
by the great physicist Helmholtz, and it has probably won 
for him more popular fame and gratitude than all his other 
work. The most remarkable thing about many inventions 
is the fact that they were not made earlier. 

124. Formulas Relating to Curved Mirrors. — In the fol- 
lowing formulas, which are here given without proof, 

D o = the distance of object-point from mirror, 
D t = the distance of image of object-point from mirror, 
F = focal length of the mirror. 

For a convex mirror we have 

A + A F 

For a concave mirror we have 

- + - = - 

A A F 

when the object-point is farther from the mirror than the 
principal focus is, and 

J_ JL_ _ 1_ 

A A ~ F 

when the object-point is between the principal focus and 
the mirror. 

It is doubtful whether work done with the cylindrical 
mirrors will be accurate enough to give results agreeing 
with these formulas. Similar formulas are used with 
respect to lenses. 



122 PHYSICS. 

QUESTIONS. 

(1) A small object is placed close to a convex mirror. 

(a) Is the image real or virtual ? 

(b) If the object is moved farther and farther away from the mirror, 
will the image at any time become real ? 

(2) If one looks at the image of his own face in a convex mirror, 
will the nose appear too prominent and the forehead and chin re- 
treating, or will the opposite be true ? 

(3) If a small object is placed close to a concave mirror — 

(a) Is the image real or virtual ? 

(&) If the object is moved farther and farther away from the mirror, 
will it reach such a position that its image will be real ? If so, what 
is that position ? 

(4) (a) Have you in using any single mirror, plane or curved, seen 
a virtual image that was inverted, as compared with the object? 

(b) Have you seen any real image, formed by a single mirror, that 
was right side up, as compared with the object ? 

(5) Do you see anything wrong with the physics of the following 
statement, copied from a prominent newspaper ? — 

There are times when the public sees things in a convex mirror, in 
which they appear broad, robust, and expanded. There are times 
when the public sees things in a concave mirror, in which they appear 
cramped, narrow, and contracted. 



CHAPTER X. 

REFRACTION OF LIGHT. 

125. Introductory. — In the experiment made with a 
prism the class may have noticed that the light did not go 
in the same direction after leading the prism as before 
entering it. Some members of the class in looking into 
pools or vessels of water may have noticed that objects 
beneath the surface are not exactly where they seem to be. 

EXPERIMENTS. 

(1) Place a straight stick in an oblique po-ition, partly in and 
partly out of water. Notice the apparent bendiDg or disconnection 
of the stick at the surface of the water. 

(2) Place on a table a pan (No. XXVIII), 15 cm. or more in diam- 
eter and with nearly vertical sides 4 or 
5 cm. high. Place a small coin on the 
bottom of the pan, and adjust the head 
in such a position that the side of the 
pan will just hide the more distant por- 
tion of the coin from the eye at E (Fig. 
85). Maintain the head in this position 
by resting it against any convenient sup- 
port ; k:ep one eye closed, and look with 
the other into the pan, just beyond the farther edge of the coin, while 
another person slowly pours in water. Have the pouring stopped 
as soon as the whole of the coin becomes visible. 

(3) Repeat experiment 2 with the eye held vertically above one 
edge of the coin, with a slender stick or a stout wire laid across the 
top of the pan, nearly in the line of vision, to serve as a point of 
departure from which to measure the apparent displacement of the 
coin, if any should be observed. 

123 




Fig. 85. 



124 PHYSIC 8. 

126. Interpretation of the Preceding Experiments. — 
Objects always appear to the eye to be in the direction from 
which the rays are travelling at the moment of entering the 
eye. Evidently, then, since the coin appeared to rise when 
the water was poured into the jar, in Experiment 2, the 
light-rays which proceeded from the coin must have been 
bent aside in some way by the water. 

In Eig. 85 the straight line CE, which passes from the 
left-hand edge of the coin C to the pupil of the eye at E, 
represents the course of a light-ray from that point before 
the water was poured into the pan. Any ray that passed 
farther to the right than CE would be intercepted by the 
side of the pan ; any ray that passed farther to the left, or 
more nearly vertical than CE, would miss the eye : hence it 
is evident that, so long as the pan is filled with air only, 
and the eye kept in the position shown, the coin cannot be 
seen. 

But as soon as water is poured into the pan, the rays no 
longer travel in straight lines from the object C to the eye. 
£ach ray suffers an abrupt change of direction at the sur- 
face of the water, and from this it follows that such rays as 
those which take the general course CS in the figure are 
finally brought to meet the eye at E. As a result of the 
bending C becomes visible, and its farther edge is seen 
apparently at 6 y ', in a position somewhat raised above the 
bottom of the pan. 

Experiment shows that the course CSE might be retraced 
by a ray. That is, a ray leaving E in the direction E8 
would reach C by the line SC. 

QUESTIONS. 

(1) If normals were drawn to the surface of the water, at the points 
about S where the rays emerge, would the bending of each ray be 
towards or from the normal (in the air) ? 



REFRACTION OF LIGMT. 



125 




(2) If the rays were passing from E to C, would the bending at 
the surface of the water be toward or from the normal (in the 

water) ? 

When we look straight clown into 
water at any small object it appears 
to be in its true direction from the eye, 
but nearer than it really is. Fig. 86 
indicates w T hy this is so. represents 
the object, E the eye, much magnified, 
and 0' the apparent position of the 
object. 

127. Angles of Incidence and Refrac- 
tion. — The change of direction which a ray of light under- 
goes when it passes obliquely from one medium into 
another is called refraction. 

The amount of the bending, or refraction, which a ray 
of light suffers at any surface depends partly upon the two 

substances which meet at this 
^ surface, and partly upon the 

angle, i (Fig. 87), which the 
ray makes with a line JYjV, 
which is at right angles with 
the surface at the point 
where the ray strikes the sur- 
face. 

If the space above the line 
AB represents the air-space, 
and that below this line the 
water, or glass, or whatever 
substance it may be that lies there, solid or liquid, the 
course of the ray is changed at the surface in such a way 
that the angle r which it makes with NN inside the solid 
or liquid is smaller than the angle ». 




C 



-B 



N 
Fig. 87. 



126 



PHYSICS. 



The angle i in Fig. 87 is called the angle of incidence. 
The angle r is called the angle of refraction. 

If the ray were represented as coming in the opposite 
direction, that is, first along R and then along /, r would 
be the angle of incidence and i would be the angle of re- 
fraction. The ray would be bent just as much at the sur- 
face as it is when going first along / and then along R. 

128. Index of Refraction, — When the direction of I is 
changed the direction of R is changed. The way in which 
the change of one depends upon the change of the other is 
easily shown by means of Fig. 88. /, /', and I" show 




three rays all of which come to the point and then sep- 
arate, the first going along i2, the second along i?', the 
third along R" . The circle whose centre is at C is drawn 
with any convenient length of radius. The dotted lines, 
w, n\ n'\ and m, m\ m" 9 are drawn from the points where 
the rays cut the circumference to the line NN 9 at right 
angles. 



BEFR ACTION OF LIGHT. 127 

If this figure has been drawn so as to acord with the 
results of experiments on light-rays, we shall have 

n __ n' _ n" 
m m! m'" 

and any one of these equal ratios is called the Index of Re- 
fraction of the second medium, heloiv AB, with reference to 
the first medium, above AB. 

n 
If now we can measure — in any given case, we shall 

have a quantity which is very useful in physics, for by 
means of it we can calculate at once the value of a new m 
to go with any new n\ that is, we can, if we know the 
index of refraction and the angle which any ray makes with 
NNvcl one medium, find without further experiment the 
angle which the same ray makes with NN in the second 

71 

medium. Exercise 20 shows how to find the ratio — for 

m 

the case of air and glass. 

EXERCISE 20. 

INDEX OF REFRACTION OF GLASS* 

Apparatus : A piece of plate glass (Xo. 28). Articles 3, 24a and 
24b. A sheet of paper and three pins. 

Place the glass, G (Fig. 89), on the paper P. Stick one pin up- 
right at the point 1 close to one of the polished edges of the glass ; 
stick the other pin at 2 close to the other polished edge. 

Look with one eye from the position S through the whole width of 
the glass at pin No. 1. Move the eye toward 3, looking all the 
time through the glass at the pin. It will presently be noticed 
that the pin seen through, the glass is not in the same direction 
from the eye as the same pin seen over the glass. That which is 

* I owe tbe plan of this admirable Exercise to Mr. F. M. Gilley of 
the Chelsea High School. It is described in Gilley's Principles of 
Physics, Allyn & Bacon, Boston. — E. H. H. 



128 



PHYSIOS. 



seen through the glass is an image of the real pin, and it is upon this 
image that the attention should be fixed. 

Continue moving the eye in the general direction of 3, keeping it, 
however, about 30 cm. from the glass, until the image of pin No. 1 
is just hidden behind pin No. 2. Then place a pin at 3, in the same 
straight line with the eye, pin No. 2, and the image of No. 1. 

Draw a fine pencil-line upon the paper close to the glass edge 
touched by pin No. 2. Then remove the glass. 

The line now drawn marks the position of the refracting surface. 
The line 1-2, Fig. 89, shows the direction, within the glass, of a 









1 


>*L G 


S 

A. 

2 




p 


\3 




Fig. 89. 



Fig. 90. 



certain ray from 1. The line 2-3 shows the course of the same ray 
after it leaves the glass at 2. The line NN' , Fig. 90, is drawn nor- 
mal to the refracting surface at the point of emergence. From this 
point equal distances are laid off, to B and to C. Lines are drawn 
from B and from G to the line i\W at right angles. 

CE-z- BD = the index of refraction from air to glass. 

EXERCISE 21. 

INDEX OF REFRACTION OF WATER. 

Apparatus : Articles 3, 14, 15, 24a, 24b, 29, 30, and a sheet of 
paper about 6 inches square. 

Put the partition i^in place, as shown in Fig. 91, and pour water 
into the jar until its surface comes within 1 or 2 mm. of the middle 
tooth of the partition. Then by means of the plunger (No. 14), at- 
tached to the side of the jar by means of its clasp, raise the level of 



BEFBACTIOjST OF LIGHT. 



129 



the water till the apparent distance between the middle tooth of the 
partition and its reflection in the water surface is less than 1 mm. 
(To see this reflection well, one should look through the wall of the 
empty part of the jar.) 

Then the brass index b is attached to the jar, as shown in Fig. 91, 
and is raised or lowered, with the tip p touching the glass, until an 
eye on the line Cg, 20 or 30 cm. from the jar, can barely see p> the 
very tip of b, apparently in a straight line with Cg. This setting 
should be made with care, and after it is made the experimenter 
must look to see whether the tooth at G is clear of the water. If its 
lowest edge touches the water the setting is useless, and all of the 
adjustments must be made anew before a reading is made. 




P 




d/-m-} 



Fig. 91. 



Fig. 92. 



When all the adjustments have been successfully made, measure 
carefully the distance from the top of the jar down to the tip, p, of 
the index, the measuring-stick being kept outside the jar. 

Measure now the inside diameter of the jar. 

Measure also, unless it is already known, the distance * of G 
below the top of the jar. 

* It is well to have this distance, which is somewhat troublesome 
to measure accurately, given by the teacher. Partitions of different 
depths might be used in order to vary the angles of incidence and 
refraction. 

If the jar used in this Exercise is not pretty level at the top, or if 
the partition is not just at the middle of the jar, it is well, after 
making one setting of the index and one measurement of its position, 
to turn the jar about, transferring the index to the other side, and 
make a new setting and a new measurement. The mean of the two 
measurements thus made should be nearly free from any error caused 
by irregularity of the jar or of the partition's position. 



130 PHYSICS. 

Now make a drawing, of full natural size, of the sides of the jar 
(inner lines), the water surface and the partition, as in Fig. 92, con- 
tinuing the partition line, by means of dots, well down into the jar. 
Put p in its proper place, and then draw the lines pC and Cg. 

Lay off Cd = Cg, and then draw the lines n and m. 

The index of refraction from air to water is — . 

m 

129. Index Different for Different Colors. — In Exercise 
20 the observer may have noticed a tint of bine or of red at 
the edge of the image of the pin. The fact is that light of 
various colors comes from the pin, and that the rays are not 
all refracted alike, the blue being refracted more than the 
red. The index of refraction is therefore different for light 
of different colors, but for our present purpose we need not 
dwell upon that fact. We get a sort of average index by 
the method of Exercises 20 and 21. 

130. Relation between Index of Refraction and Velocity 
of Light. — The velocity of light in any transparent sub- 
stance depends on the nature of the substance. It is 




greatest in a so-called vacuum. It is least in the most 
highly refractive substances, and, indeed, the index of re- 
fraction for any given substance depends upon the rate at 
which light travels through it. 

This is sometimes illustrated by an analogy suggested by 



REFRACTION OF LIGHT. 



131 



the march of troops over ground of various kinds. Suppose 
a column of troops to be marching over smooth ground, 
represented by the space to the left of the line SS ' in Fig. 
93. The front of the column being at AB, let the line 
SS' represent the border of a marsh or other difficult ground. 
Upon entering, the right of the column, B, first encounters 
the marsh, and the soldiers at B will fail behind those of 
the rest of the front. In consequence of this the column 
will, one part after another, wheel to the right until, when 
the whole front has entered the marsh, it will have the new 
direction shown by the line A'B'. Substitute for the 
column of troops a beam of light, and for the marsh a 
highly refractive transparent substance, and one may get 
some notion as to how refraction depends upon the retard- 
ing effect of refractive substances upon light-rays. 

131. Total Internal Reflection: Critical Angle. —In 

Fig. 94 we have air above the horizontal line and w^ater, 




Fig. 94. 

glass, or some such transparent medium below the line. A 
ray of light R x may come from beneath to the surface at 
such an angle with the normal that it will after refraction 
at be parallel to the refracting surface. The ray R t com- 
ing up to at a larger angle with the normal will not pass 
out to the air, nor will it skim alons; the surface. It will 



132 



PHYSICS. 



be reflected at the point 0, the surface acting as a perfect 
mirror, and will follow the course R'^ the angle of reflec- 
tion being equal to the angle of incidence. 

The angle a, ivhich must not be exceeded if the ray is to 
2mss ont into the air, is called the Critical Angle. 

The reflection which takes place when this angle is ex- 
ceeded is so good that it bears the especial name total 
reflection. 

EXPERIMENTS WITH TOTAL REFLECTION. 

(1) With the eye at E, Fig, 95. look at right angles into a glass 

prism shaped like ABC, at the same 
time holding an object at 0. Note the 
position of the image 0' and its re- 
markable distinctness. 

(2) In Fig. 96 SS' is a disk or square 

of thin wood about 10 cm. wide, 

LO is a piece of knitting-needle about 

8 cm. long. The wood floats in water 

which fills a vessel to the brim AB 

Push the needle down until its upper end is nearly level with the 

upper surface of the board, and look down obliquely through the 

water, close past the margin of the board, at the lower extremity of 



JE± 





Fig. 96. 

the needle. Now draw the needle up, little by little, through the 
floating board until the point is reached at which the needle just 
vanishes from view, the line of sight being made at last as nearly 
horizontal as possible. Lift the board from the water, and note how 
much of the needle still projects below the board. 



REFRACTION OF LIGHT, 



133 



When the point is at 0', the light-ray going from it to S passes out 
into the air. When the point is at 0, a light-ray OS suffers total 
reflection along SR. The angle OSN 1 , or its equal SOL, is nearly 
equal to the critical angle. Of course no great accuracy can be 
expected here. 

Effect of Transparent Plates and Prisms. 

132, Transparent Plates. — A plate of glass, or other 
transparent material, with plane parallel 
sides, as in Fig. 97, refracts light which 
enters it obliquely, but refracts it equally and 
in the opposite direction when it comes out 
at the opposite side of the glass, so that the 
entering and emerging rays are parallel to 
each other, although, as Fig. 97 shows, they 
do not lie in one straight line. 

Evidently a thick plate of glass will, other 
things being equal, set the emergent ray 
farther to one side, from the line of the original ray, than 
a thin plate will. 

133. Prisms. — A. prism, in the study of light, is usually 
a piece of glass, or other transparent material, bounded by 
three rectangular and two triangular faces. DEF in Fig. 
98 represents one end of such a prism. 

J) 




Fig. 97. 




It is evident that light entering the face DE from air will 
be refracted toward the normal NM. Going through the 
prism to the face DF 'it passes out into the air, being re- 



134 PHYSICS. 

fracted again, this time from the normal M'N\ so that the 
two refractions have bent the ray far from its original 
direction. 

The total bending or deviation suffered by a ray in pass- 
ing completely through a prism depends on a number of 
things. 

1st. On the angle which the two faces passed through make 
with each other. This angle is called the refracting angle : 
see D in Fig. 98. 

The greater this angle is, other things being equal, the 
greater the total deflection will be. We have seen in § 132 
that if the two faces are parallel the total deviation is zero. 

2d. On the color of the ray. 

This fact has already been noticed. Eed light is deviated 
less than blue light. 

3d. On the angle tvhich the ray makes with the first sur- 
face. 

The total deviation is least when the ray strikes in such 
a way as to follow, within the prism, a course parallel to 
ZE7, ."Fig. 99, which makes the distance AI equal the dis- 
tance AE, and makes the refraction equally great at both 
surfaces. 

EXPERIMENT. 

Repeat the experiment of § 89, varying the angle at which the 
sunlight strikes the first face, in order to show that there is one in- 

A 




clination which gives a less total deflection of the light than any 
other position. 



REFRA CTION OF LIGHT. 135 

4th. On the material of the prism. 

All kinds of glass do not refract equally. 

134. Dispersion: The Spectrum. — The separation of rays 
of different colors by a prism is called dispersion. 

The spot or band of colored light produced by the dis- 
persion of a sunbeam is called the solar spectrum. 

It is customary to divide the spectrum into seven regions, 
called red, orange, yelloiu, green, Hue, indigo, violet, and to 
call the general colors of these the primary colors, to dis- 
tinguish them from those formed by compounding two or 
more of them. This division of the spectrum is a mere 
matter of convenience. We might name a hundred colors 
of the spectrum if we chose to do so. 

So long as we keep to any one refracting material the 
dispersion is, in general, greater when the average deviation 
of all the rays is greater. Thus with a given prism the dis- 
persion is least when all the rays go through the prism as 
the ray IE goes in Fig. 99. 

When prisms of different material are used, two kinds of 
glass for example, one may disperse the rays more than 
the other, while producing no greater average deviation of 
all the rays ; or one may disperse the rays about as much as 
the other while deviating them, as a whole, much less. 

135. " Achromatic " Prisms. — Two prisms of nearly equal 
dispersive power but of unequal deviating power 
may be combined, as in Fig. 100, making a com- 
pound prism which produces considerable deviation 
with very little final dispersion. Such a combina- 
tion is called achromatic, that is, colorless. 

Achromatic combinations of lenses (§ 149) are 
used in many optical instruments. Fig. ioo. 




136 



PHYSICS. 



Lenses. 

136. Shapes of Lenses. — A lens is, usually, a piece of 
glass whose two faces are parts of spherical surfaces. 

Sometimes there is a cylindrical surface between the two 
spherical faces. 

Fig. 101 shows various lenses as they would look if cut 
through the middle. 




Fig. 101. 



Lenses are classed as convex, or converging, and concave, 
or diverging. Convex lenses are all thicker in the middle 
than at the margin, and cause parallel light-rays to con- 
verge, as in Fig. 102. Concave lenses are thinner in the 




Fig. 102. Fig. 103. 

middle than at the margin, and cause parallel light-rays to 
diverge, as in Fig. 103. 

Some of the lenses used in the most accurate optical in- 
struments have convex or concave surfaces, which are not 
strictly parts of spherical surfaces. Such lenses possess 
certain advantages over spherical-surface lenses (see § 147). 

137. Definitions Relating to Lenses. — The lenses we 
shall use will be much like No. 1 in Fig. 101. The two 
sides are supposed to be just alike. 



REFRACTION OF LIGHT. 



137 



To understand such a lens better we will make use of 
Fig. 104. 

(7 is the centre of the spherical surface of which A SB is 
a part. It is called the centre of curvature of the face 
ASB. C is the centre of curvature of the face ABB. 




The straight line HCOO'K, continued to any distance in 
each direction, is called the principal axis of the lens. 

Any straight line going, like LM, obliquely through the 
centre of the lens is called a secondary axis of the lens. 

If the two faces of a lens are exactly alike, as we suppose 
them to be here, any ray of light going through the centre 
of the lens, the point 0, will hare the same direction after 
leaving the lens as before entering it, because the two little- 
spots of surface at which it enters and leaves the lens are 
parallel to each other, so that the ray is affected just as if 
it were going through a plate with parallel faces.* is 
called the optical centre of the lens. 

Eays entering a convex lens "parallel to its principal axis, 
as in Fig. 102, are refracted in such a way that after leav- 
ing the lens they will cross this axis. They do not all cross 
at one point, but if the faces are near together, and are very 
small parts of spherical surfaces, as in our lenses, such rays 
will cross at or near a certain point, F, on the principal 

* The direction of the ray within the lens is, of course, not quite 
the same as its direction before entering. This fact is not shown in 
Fig. 104. 



138 



PHYSICS. 



axis, and this point is called the principal focus of the 
lens. There are two principal foci, one on each side of the 
lens. See points i^and F' in Fig. 104. 

The distance from the principal focus to the nearer face * 
of the lens is called the focal length of the lens. 

Focal length is a quantity of very great importance in 
dealing with lenses, and the next Exercise will show how 
to find it by experiment. For this purpose we need to have 
the light come to the lens in rays nearly parallel to each 
other and to the principal axis. This we can do by taking 
the light from any small spot of any distant but distinct 
object; for instance, a chimney or a church-spire outlined 
against the sky. 

EXERCISE 22. 
FOCAL LENGTH OF A CONVERGING LENS. 

Apparatus : The lens (No. 31) mounted on a block. A meter-rod 
(No. 2). A small block (No. 21) bearing a white cardboard screen 
(No. 32). A common pin. 




Fig. 105. 

First Method. — Place the lens and the screen upon the rod, as in 

Fig. 105, and point the rod at some distant object, seen against the 

sky, in such a way that the light from this object will pass from the 

lens and then fall upon the screen. Move the screen back and forth 

* Bee Appendix I, 



REFRACTION OF LIGHT. 139 

until that part of the image * which lies on or near the principal axis 
of the lens is made as distinct as possible. Then by means of the grad- 
uations of the meter-rod, or by an independent measuring-stick if this 
is preferred, note the distance from this part of the image to the 
nearer face of the lens. This is the focal length. 

Second Method. — Remove the screen from its block and put the 
pin upright in its place. Let the pin, thus mounted, be placed on 
the meter-rod, about as far from the end of the rod as the pupil 
usually holds a book from his eyes when reading. Place the lens 
somewhat farther from the same end of the rod. 

Place the eye at this end of the rod and, looking sharply at thepm, 
direct the rod and adjust the lens in such a way that the light from 
some distant object will pass through the lens and form an image in 
the air close to the pin. To decide whether the image is nearer the 
eye than the pin is, move the eye to and fro, to the right and the left, 
watching the pin and the image. f If the pin is more distant than 
the image, it will, when the eye is moved toward the right, appear 
to move across the image toward the right. If the pin is nearer than 
the image, it will, when the eye is moved toward the right, appear 
to move across the image toward the left. The rod should not be 
held in the hands during this test, but should be placed on some 
steady support. 

Continue the adjustments until the test described fails to show 
which of the two, the pin or the image, is nearer the eye. Then meas- 
ure the distance from the pin to the lens. It should be the focal 
length of the lens. 

Compare the values of the focal length given by the two methods. 
The second method is more difficult, but it is instructive, and it 

*The image is formed because light coming from anyone small 
spot of the object is brought to a small spot again by the lens. The 
image is made up of such small spots each in its own place. For the 
purposes of this Exercise the distant object need not be more than 30 
or 40 feet from the experimenter. The images on the screen will 
be much more distinct if the apparatus is used in the back part of 
the room, well away from the windows. 

f To see the reason of the test just described, close one eye and hold 
the two forefingers, some inches apart, in line with the other eye, so 
that one finger hides the other. Then move the eye to the right 
and left, and notice the apparent movement of the fingers with 
respect to each other. 



140 PHYSICS. 

can be used in cases where the image is too faint to show clearly upon 
the screen. 

138. Discussion of Exercise 22. — It is common to speak 
of the rays coming to a lens from a distant object as parallel 
rays. . This does not mean that rays coming from different 
parts of the object to the lens are parallel to each other. It 
means merely that rays coming from any one spot of the 
object to the lens are parallel, or very nearly parallel, to 
each other. In fact, if rays from the different parts of a 
luminous body could be converged to the same point, the 
result would not be an image repeating the features of the 
original objects. It would be a mere point, or very small 
patch of light. 

The image seen in the Second Method is, like that of the 
First Method i a real image (§ 109), but it is in the air. 

As there is an image in the air, we may well inquire why 
this image cannot be seen by a whole class at once without 
the use of a screen. It is because the light forming the 
image in the air goes straight on through this image, and 
can be received only by placing one's self behind the image. 
The light which forms an image upon a screen is by the 
threads of the screen reflected back in all directions, and 
therefore some part of it reaches every eye. 

QUESTION. 

If a bright point were placed at the principal focus of a lens, what 
direction would the rays going from this point to the lens have after 
passing through the lens ? 

139. Object-distance and Image-distance: Conjugate 
Foci. — Two points so placed tvith respect to a lens that an 
object placed at either of them will have an linage at the 
other are called Conjugate Foci of the lens. 



REFRACTION OF LIGHT. 141 



EXERCISE 23.* 

RELATION OF IMAGE-DISTANCE TO OBJECT-DISTANCE : 
CONJUGATE FOCI OF A LENS. 

Apparatus : The same lens that was used in Exercise 22. A 
meter-rod. Block (No. 9). Small block (No. 21), with a cardboard 
screen (No. 32). Small kerosene lamp with an asbestos band around 
the chimney (No. 33). 

Arrange the apparatus according to Fig. 106. The hole in the as- 




Fig. 106. 

bestos band, lighted up by the name behind, is the object the image 
of which is to be received upon the screen. One end of the meter- 
rod is placed vertically beneath this illuminated hole. 

Place the screen at first at a distance from the object about equal 
to three times the focal length of the lens. Then move the lens back 
and forth on the rod between the object and the screen, and see 
whether in any position it gives upon the screen a clear image of the 
object. If it does, measure the distance from the lens in this position 
to the object, and write this distance as the first number in a record- 
column headed D (object-distance). Measure also the distance from 
the lens to the screen, and put this distance as the first number in a 
record-column headed Di (image-distance). 

If, with the present position of the screen and object, there is no 
position of the lens that will cause a distinct image of the object 
to fall upon the screen, move the screen one or two centimeters far- 
ther from the object, and then try again to get a good image. If still 
none is found, move the screen still farther away, continuing the 
trial till a distinct image is obtained. Then measure and record the 

* To economize space upon the laboratory- tables it will probably 
be necessary to have pupils work in pairs in this Exercise. Each pair 
should know the focal length of its lens at the outset, so as to lose no 
time in beginning the Exercise. 



142 PHYSICS. 

D and Di as already described. (Very little time need be spent upon 
these first successive trials.) 

Then at one move place the screen about 10 cm. farther still from 
the object, find a position of the lens that will give a distinct image, 
measure and record D Q and Di as before. Without moving the screen, 
see whether there is any other position of the lens that will give a 
distinct image ; if there is, measure and record the D Q and the Di for 
this position of the lens. 

Move the screen 10 cm. farther away, and then do exactly as 
before. 

If there is time, move the screen two or three more times, adjust- 
ing the lens, measuring, and recording each time. It is better to 
make a moderate number of settings and readings well than a large 
number carelessly, but an error of one or two millimeters in these 
readings will be of no great consequence. 

140. Discussion of Exercise 23. — The distance from 
object to image in any case of Exercise 23 is D Q + D { , and 
we may call this D oi . This distance was shortest in the 
first case recorded. Let each member of the class divide 
the D oi of this case by the focal length of his lens. Is there 
any general agreement between the quotients thus found ? 

When the screen was farther away, was there usually 
more than one position of the lens that would give a distinct 
image, the screen remaining unmoved ? 

If you were told that in a given case the D Q was 20 cm. 
and the A 60 cm., could you tell what the other possible 
D Q and Di would be for the same positions of object and 
screen ? Look at your record-columns for Exercise 23, and 
see whether they help you to answer this question. 

Let each member of the class call F the focal length of 
the lens which he used, and let him test the truth of the 
formula. 

1 = 1 + 1 



REFRACTION OF LIGHT. 143 

or, what means the same, 

D xD =F(D +A), 
for all cases tried and recorded Ly himself in Exercise 23. 

PROBLEMS. 

(1) Do for a certain case is 50 cm. and Dx is 100 cm. How great is 
Ft 

(2) If Di is 80 cm. and Fis 20 cm., how great is D ? 

(3) If D Q =Di, we will call each D. 

(a) What in this case is the relation between F and D1 

(b) How does this agree with your observations in Exercise 23? 

141. Real Image Formed by a Lens. — In the preceding 
Exercises the object presented to the lens has been small, 
or has been at such a distance as to give a rather small 
image. It is now desirable to study larger images, and to 
study them with especial reference to their shape and size, 
rather than their distance from the lens. We shall in the 
next Exercise find the shape and size of an image of an 
arrow placed at right angles with the principal axis of the 
lens and not far from the lens. We shall not attempt to 
find the whole image at once, but shall find separately the 
images of several points of the arrow, and then make an 
approximate image of the arrow by connecting these points. 

EXERCISE 24. 

SHAPE AND SIZE OF A REAL IMAGE FORMED BY A LENS. 

Apparatus: The lens (No. 31). Measuring-stick (No. 3). Block 
(Xo. 21) carrying in the narrow slot on its top a piece of wire (No. 
34) extending first horizontally and then downward (see Fig. 108). 
A ruler (No. 24). Block (No. 25). A sheet of paper about 30 cm. 
wide and 1 m. long, having near one end an arrow 8 cm long, drawn 
at right angles with a pencil-mark about 30 cm. long, and marked, or 
numbered, as shown by Fig. 107. Weights (No. 19) to hold the cor- 
ners of this sheet in place on the table. 

Arrange the apparatus as shown by Fig. 108, the centre of the lens 
over a point on the long pencil-mark, at a distance from the centre of 



144 



PHYSICS. 



the arrow about equal to one and a half times the focal length of the 
lens, and block Xo. 25 in such a position that the vertical mark upon 
its face points straight down to point Xo. 3 of the arrow. This ver- 
tical mark will now cross the principal axis of the lens, if the lens is 
accurately placed. 

Place the other block near the other end of the paper in such po- 
sition that the vertical part of the wire it carries shall be near the 



A1 



■m 



Fig. 107. Fig. 108. 

principal axis of the lens. Keep the eye 20 or 30 cm. distant from 
this part of the wire, on a level with the centre of the lens and in line 
with the centre of the lens and the vertical part of the wire. Look 
at this part of the wire so as to see it distinctly, and note whether 
you can see at the same time, near the wire, the image of the pencil- 
mark on the farther block. If so, find out by moving the eye to the 
right or left, as in Exercise 23, whether this image is more or less 
distant from the eye than the vertical wire is. Then move the block 
carrying the wire into such a position that the image and the wire 
seem to keep close together when the eye is moved a considerable 
distance to the right or left. When this adjustment is made, put a 
dot on the paper just beneath the vertical wire and mark this dot 3. 
It represents the image of object-point Xo. 3. 

Find in a similar manner the image-points 1, 2, 4, 5, corresponding 
to the object -points 1,2, 4, 5. The experimenter must take care not 
to let any idea he may have as to the position where an image-point 
ought to be affect his judgment in deciding where it is. 

After all the five image-points are found, connect them, No. 1 to 
No. 2, No. 2 to Xo. 3, etc., by means of straight lines, thus getting 
a rough representation of the whole image. 



REFRACTION OF LIGHT. 145 

Draw from each object-point toward the corresponding image-point 
a straight line as long as the ruler (No. 23), and note the point where 
^hese lines cross each other. 

142. Formation of the Image in Exercise 24. — The for- 
mation of the image-points in Exercise 24 is illustrated by 
Fig. 109. One ray from the object-point A follows a 
secondary axis (§ 137) passing through the centre of the 
lens, and its direction after leaving the lens is the same as 
before entering it. (Its direction inside the lens is not 
quite the same, but the figure does not show this.) 

Another ray from A runs parallel to the principal axis 
(§ 137) before entering the lens, and will therefore pass 




Fig. 109. 

through the principal focus, /, on the farther side of the 
lens. The crossing of these two rays at A' shows the posi- 
tion of the image of A. 

In a similar way B\ the image of i?, is located. 

143. Size and Shape of Image. — If a straight line is 
drawn from A' to B' in Fig. 109, we mav call this the 
length of the image, although the images of points between 
A and B will not lie on this line. It is evident from Fig. 
109, and also from the figure obtained in Exercise 21, that 
the distance A'B' is to the distance AB as the distance of 
A'B' from the lens is to the distance of AB from the lens. 

The curved shape of the image obtained in Exercise 
24, if the work has been correctly done, is due to the 
fact that the ends of the object-arrow are farther from the 
lens than the centre of the arrow, and to the further fact 



146 PHYSIOS. 

that the focal length along a secondary axis is less than the 
focal length along the principal axis. This latter fact can 
easily be shown by direct experiment with either method 
of Exercise 22. 

144. Virtual Image Formed by a Lens. — We see in 

Exercise 24 and in Fig. 109, where the object-point is 
farther from a lens than its principal focus is ; that the rays 
going from this object-point to the lens are bent by the lens 
in such a way that, after leaving it, they converge to a point 
again. We know, too, that if the object-point were placed 
at the principal focus the rays going from it to the lens 
would emerge from the lens parallel to each other. 

It is not difficult to see that, if the object-point were 
placed letiueen the lens and its principal focus, the rays 
going from it to the lens would be divergent still, after 
leaving the lens, though less divergent than before entering 
it. In the next Exercise we shall have a case of this kind. 

EXERCISE 25, 

VIRTUAL IMAGE FORMED BY A LENS. 

Apparatus: The same as for the preceding Exercise except that 
the sheet of paper need not be more than one half as long, and that 
the arrow upon it should be 4 cm. long and about 20 cm. distant from 
one end. 

Place the lens between the arrow and the nearer end of the sheet 
of paper, at a distance from the arrow equal to about two-thirds of 
its focal length, and in such a position that its principal axis extends 
over the middle point of the arrow. Place the small block (No. 25) 
with vertical pencil-mark pointing straight down at the middle 
point, No, 3. of the arrow. Turn the vertical part of the wire on the 
other block so that it will point up instead of down, and place this 
block some distance behind the other one. 

Holding the eye 20 or 30 cm. from the lens, look through the lens 
at the image of the vertical pencil-mark, and at the same time oxer 
the lens at the vertical part of the wire. Bring the wire into line 
with the image, and then by the usual test find which of them is the 
more distant. Move the wire back and forth until it coincides in 



REFRACTION OF LIGHT. 147 

position with the image. Then mark with a figure 3 the point just 
under the vertica lpart of the wire. This represents the image of 
object-point No. 3. 

In a similar manner locate the images of points 1, 2, 4, and 5. 

Connect the image- points by straight lines, from 1 to 2, from 2 to 
3, etc.. thus forming an image of the arrow. 

Draw a straight line from each image point to its corresponding 
object-point, and note where these lines will cross each other if con- 
tinued. 

145, Formation of the Image in Exercise 25. — The 

images observed in Exercise 25 were virtual images. They 
could not be shown tipon a screen, and were not formed by 
the actual crossing of light rays. Fig, 110 will serve to 
illustrate the way in which virtual image-points are formed. 

Let AB be the object, placed between the lens LL' and 
the principal focus F\ To find the position of the virtual 
image of the point A, draw ^[/parallel tc the principal axis 
of the lens, This ray will, after leaving the lens, pass 
toward F, the principal focus * on the farther side, and so 
will appear to have come along the path 31 F. 

Draw another ray, AC\ passing through the centre of the 
lens This ray will, after leaving the lens, have the same 
direction as before entering it, and will be represented by 
the line CiV". If, then, we carry back the line CiVtill it 
crosses the line MF, also carried backward, the point A\ 
where the crossing occurs, is a point from which both of 
the rays appear to come. A' is, then, the virtual image 
of A. 

By a similar process B' is found to be the virtual image 
of B. 

P\ the image of the point P, is here represented as lying 
in the straight line between A' and B'. It is usually so 

* The dotted lines drawn from M and N to F 'in Fig 110 are not 
intended to show the actual course of the rays within the eye. 



148 PHYSICS. 

represented in books. Exercise 25 shows that it does not 
lie there, 

The image A'B' is evidently larger than the object AB. 
Whenever a virtual image is forced by a convex lens, this 
image appears, to an eye placed in any ordinary position on 
the other side of the lens, larger than the'real object would 

^ A' 




i -^^— "" 

Fig. 110. 

look if held at a comfortable seeing-distance from the eye. 
Hence the name magnifying-glass^ so commonly given to a 
lens used as in Fig. 110. 

146. Application of Formula. — The formula used in 
§ 140 to express the relation betweeen focal length, object- 
distance, and image-distance in the case of real images, can 
be adapted to use with virtual images by merely changing 
the sign of one term, so as to make 

i A A 

To illustrate the use of this formula it will be well to 
measure the distance from lens to object-point 3, and from 
lens to image-point 3, in the diagram made in Exercise 25, 
and try them in the formula, with the known value of F, 



REFRACTION OF LIGHT. 149 

147. Spherical Aberration in Lenses. — All the rays going 
from a point to a lens A do not, after passing through 
the lens, converge to a single point 7. Those which go 
through the lens near its margin converge to a nearer point 
J\ This imperfection of a lens is called spherical aberra- 
tion. 

When a very clear-cut image is needed, it is customary 
to put a stop in front of the lens; that is, a thin metal 




plate with a hole which permits only those rays to pass 
which are near the principal axis of the lens. 

Lenses can be so constructed, with surfaces not quite 
spherical, as to do away with this defect in great part, for 
light of any one color, but such lenses are difficult to make 
and are uncommon except in large telescopes. 

148. Chromatic Aberration in Lenses. — Ordinary lenses, 
made of a single piece of glass, give rise to colored fringes 
or borders about the images which they produce. The 
cause for this defect, which is called chromatic aberration, 
is this, that the objects looked at send more than one kind 
of light to the lens and that rays of different colors are 
not refracted equally by the lens, and so do not come to a 
focus equally near the lens. 

But little trouble from this source is experienced in the 
use of lenses of slight convexity, whose images are not to be 
further magnified; as, for instance, in spectacles and ordi- 
nary magnify ing-glasses. " Stopping out " the greater 



150 PHYSICS. 

portion of the surface of a lens with a circular diaphragm, 
which allows light to pass only through a small portion of 
the lens near its centre, improves its performance greatly. 
How much help such diaphragms give by reducing spherical 
and chromatic aberration, may be learned by taking out 
some or all of the diaphragms of an ordinary cheap spy- 
glass, and then looking with it at distant objects in bright 
sunlight. 

149. Achromatic Lenses.— Fortunately for the manufac- 
turers and users of optical instruments, it is possible to 
make an achromatic lens, or one, at any rate, 
which is practically achromatic. This is usually 
|b accomplished by uniting into one lens two sepa- 
rate lenses,* one, A, of flint-glass, and the other, 
B, of crown-glass, as shown in Fig. 112. A con- 
vex lens made in this way has, on the whole, a 
converging effect on parallel rays, while at the 
same time the superior dispersive power (§ 134) of 
the flint-glass enables the lens A, though of less 
fig. 112. refractive power than the lens i?, just to coun- 
teract the dispersive tendency of the latter. Many of the 
lenses used in optical instruments of the best quality are 
achromatic. Eye-pieces (§ 164), however, of the ordinary 
pattern do not require achromatic lenses. 

A large lens practically free from spherical and chromatic 
aberration is a marvel of skillful and patient work. Grlass 
suitable for making a large lens of the best quality is very 
difficult to procure, as a very slight flaw or unevenness of 
quality may spoil a large block. The shaping and polish- 
ing and testing of the largest lenses, after the proper kind 
of glass is obtained, is a work of years, and men who are 

* Sometimes more than two pieces are employed in making an 
achromatic lens. 




REFRACTION OF LIGHT. 151 

skillful and patient enough to do it become known through- 
out the world. 

For many years the largest and best lenses for great 
telescopes have been made by Alvan Clark and his two sons 
of Cambridge, Massachusetts; but now all of these famous 
men are dead. The largest lenses ever made, 40 inches in 
width, were placed in the great telescope of the Observa- 
tory of Chicago University by the last of the Clarks a few 
weeks before his death in 1897. 

QUESTIONS AND PROBLEMS. 

(1) An object is placed at a great distance from a converging lens 
and on its principal axis. 

(a) What changes of position will the image of this object undergo 
while the object is moved along the principal axis up to the surface 
of the lens ? 

(b) In what part of this operation will the image be erect and in 
what part inverted ? 

(c) In what part will it be real and in what part virtual ? 

(2) In Exercise 25 the .virtual image of a straight line was found 
to be a curve. How should a line be curved with respect to the lens 
in order to make its virtual image a straight line ? 

(3) The focal length of a certain convex lens is 15 cm. 

(a) How far from the lens will the image be if the object is 30 
cm. from the lens ? 

(b) How far if the object is 10 cm. from the lens ? 

(4) An object is 40 cm. from a convex lens and the image equally 
far from the lens. What is the focal length of the lens ? 

(5) If the object mentioned in problem 3 is 5 cm. long, how long 
will each of the images there mentioned be ? (In answering this 
question disregard the curvature of the images.) 

(6) A bright point, which is more distant from a converging lens 
than its principal focus is, sends white light to the lens. Which falls 
nearer the lens, the red image of the point or the blue image ? Why ? 

(7) What would be the answer to the questions in (6) if the point 
were between the lens and its principal focus? 




CHAPTEE XI. 

THE EYE: SIGHT AND COLOR. 

150. Parts of the Eye. — The eye as an optical instru- 
ment consists of a liquid lens A (Fig. 
113) called the aqueous humor, a solid 
lens, B, called the crystalline lens, a 
transparent jelly-like mass 0, called 
the vitreous humor, and a screen rr, 
called the retina, upon which the 
image of the object looked at falls. 

The aperture at the back of the 
FlG ' 118 ' eye is occupied by the optic nerve 

leading from the retina to the brain. 

151. Accommodation. — Muscles attached to the lens B 
have power to change its form to some extent, thus adapt- 
ing the eye to see distinctly near or distant objects at will. 
This is called the power of accommodation. 

A normal eye, that is, an eye approved by physicians, has 
such shape as to give upon the retina distinct images of 
very distant objects without effort. In accommodating 
itself to see nearer objects such an eye has to make an effort, 
which grows greater as the distance lessens, but does not 
become painful until the object looked at is less than eight 
or ten inches from the eye. 

152. Far-sight and Near-sight. — Some eyes lack the 
power of accommodation for near objects, and are called 
far-sighted, or long-sighted, although they cannot see dis- 
tant objects any better than normal eyes can. 

152 



THE EYE: SIGHT AND COLOR. 153 

Some eyes are slightly egg-shaped, the retina being 
farther back than in normal eyes. These eyes are called 
near-sighted, or short-sighted, because they are well adapted 
for seeing near objects, while they cannot see distant objects 
distinctly. 

For some purposes near-sighted eyes have a certain 
advantage over normal eyes, for they enable their possessor 
to hold an object very near, when there is need, and so 
make it look larger than it would look to the normal eye. 

153. Eye-glasses.' — Ear-sighted eyes must wear convex 
lenses to help them converge the rays from a near object to 
an image upon the retina. Near-sighted eyes must wear 
concave lenses to prevent the rays sent by a distant object 
from coming to an image in front of the retina. 

The Perception of Color. 

154. The Color-sense. — In the retina are found the ends 
of the nerves through which we get the sensation of light 
and of color. 

Although the eye can distinguish scores of different tints, 
it is believed that the sets of nerves operating in the percep- 
tion of colors are very few, probably not more than three 
or four. Each set of nerves is supposed to give one peculiar 
color sensation and only one ; but the combination of these 
few primary color sensations in various proportions is sup- 
posed to give all the other color sensations. 

It is very generally believed that the primary color sensa- 
tions are three — red, green, and violet. 

155. Mixing Color Impressions. — The most convenient 
way to find the effect of mixing color sensations is to place 
variously colored pieces of paper on some body which can 
be made to spin rapidly before the observer's eyes. Tops 
or other whirling apparatus No. XXXV, for example, can 



154 PHYSICS. 

be used for this purpose, and indeed the whole outfit for 
this kind of experimentation is now readily obtained. 

EXPERIMENT. 

Place a red paper, a green paper, and a violet paper upon a whirl- 
ing apparatus, and so vary the proportions of the visible parts of 
these papers that when rapidly whirled before the eye they will 
produce the effect of gray. (In the study of color all shades of gray, 
from brilliant white to dead black, must be classed together as white, 
the difference between them being merely a difference of brightness.) 

156. Complementary Colors. — It has already been shown 
that ordinary white light is composed of many different 
colors, ranging from red to violet, but it is not necessary to 
put together all of these colors in order to get the sensation 
of white. There are many pairs of colors, any one pair of 
which will give the sensation of white when its elements 
are mixed in the right proportions. The two colors making 
such a pair are called complementary to each other. Thus, 
according to Eood, 

red is complementary to green-blue, 

orange " " " cyan-blue (between blue-green 

and blue), 
yellow " " " ultramarine-blue, 

greenish-yellow " " violet, 

green te " purple. 

EXPERIMENT. 

Place blue and yellow disks upon the whirling apparatus, and so 
proportion the visible parts that when revolving rapidly they will 
produce the effect of gray. 

Try the same experiment with other pairs of complementary 
colors. 

157. Fatigue of the Retina. — If one looks steadily for a 
short time at some strongly colored object held against a 
background of gray or white, that spot of the retina upon 
which the image of the colored object falls loses in part, for 



THE EYE: SIGHT AND CO LOB. 155 

the time being, the power of giving the particular color 
sensation which it is furnishing, while its power of giving 
other color sensations may remain as great as ever. This 
temporary loss of power is called fatigue of the retina, and 
it may give rise to curious effects. 

EXPERIMENT. 

Hold a piece of bright green paper against a white background, 
and look very steadily at one spot on this paper for thirty seconds. 
Then look steadily at some one spot of the white surface for a few 
seconds and note any peculiar color effect that is observed. The 
color complementary to green will probably appear as a patch upon 
the white, the shape of this patch being exactly like that of the 
green paper. 

Try the same experiment with other colors. 

158. After-images. — The effects observed in the follow- 
ing experiment are still more curious than those of § 157. 

EXPERIMENT. 

Look steadily for half a minute at some 'particular spot on a win- 
dow haying the sky as a background. Then close the eyes and wait 
a few seconds for the figure on the window to show out against the 
darkness. Watch the changes of color the figure undergoes. Ob- 
serye that details appear in this persisting image which were not 
noticed while the eyes were open. 

11 After-images " like the one here mentioned, cannot be 
the work of memory. They must be due to some change 
of state in the retina, some real impression made there, 
which lasts for a considerable time but gradually passes 

away. 



CHAPTER XII. 

OPTICAL INSTRUMENTS. 

159. Importance of Optical Instruments. — Much of the 
progress of science during the nineteenth century has been 
due to improvements in the construction of optical instru- 
ments and their more general use in scientific investigations. 

Improvements in telescopes and the invention and per- 
fection of the spectroscope have enabled the astronomer to 
discover, and even to measure, objects and motions whose 
existence was unsuspected by the observers of two genera- 
tions ago. The chemist is to-day able by means of the 
spectroscope to ascertain in a few minutes the presence, in 
a substance of unknown composition, of elements which it 
would have taken him days to detect by purely chemical 
means. 

To the physician, the food-analyst, the manufacturing 
druggist, and to those engaged in many other professional 
or technical occupations, the microscope is a necessary piece 
of apparatus, a tool of daily, almost hourly, use. 

Optical instruments comprise a great variety of combina- 
tions of mirrors, lenses, and prisms. Only some of the 
simpler ones can be referred to in an elementary book on 
physics. 

160. The Photographer's Camera. — This instrument 
consists essentially of a box, in the front of which is fastened 
a convex lens or a combination of lenses, L (Fig. 114), the 
distance of which from a ground-glass screen, P, at the 

156 



OPTICAL INSTRUMENTS. 



157 



other end of the box. may be varied at will. An inverted 

and usually diminished real image of any o.itside object not 
too near L may be formed on P. When this adjustment 
lias been precisely made, the lenses are covered with an 
opaque cap; a plate of ordinary glass, coated with a film of 
gelatine made sensitive to light by the presence in it of cer- 
tain compounds, usually of silver, is substituted for P; the 
cap is then- removed, and the light is allowed to act for a 













A 


p 




L 




\/ \J 











Fig. 114. 

sufficient time upon the sensitive plate, after which the cap 
is replaced and the plate removed and " developed " into a 
photographic t; negative/* 

Those who are interested in practical photography will 
find in Exercise *24 some explanation of the difficulty ex- 
perienced in making all parts of the ground-glass screen 
show clear images at the same time; and in § 1-47 there is 
a suggestion as to the effect of ; * diaphragms " with larger 
or smaller holes. 

161. The Magic-lantern. — This instrument, known also 
by various other names, stereapticon, for instance, requires 
a powerful source of light, such as a large kerosene-flame, 
or some form of calcium-light A (Fig. 115), in which a 
cylinder of quicklime is heated by a flame formed by burn- 
ing together oxygen ami coal-gas, or. bust of all. the electric 
arc-light. By means of a large lens B (Kg. 115), called 



158 



PHYSICS. 



the condenser, a powerful beam of light from this source 
is thrown upon the painted or photographed " slide," the 




Fig. 115. 

image of which is to be exhibited. This slide is pushed 
into the opening (7, a little outside the focus of a smaller 
convex lens or a pair of such lenses, D, and a greatly 
enlarged real image of the slide is thrown upon the screen. 

The throwing of large images upon a screen is called pro- 
jection of these images and apparatus used for this purpose 
is called projecting apparatus. 

162. Projecting a Spectrum. — A kind of spectrum has 
been shown in the experiment of § 89, but a better disper- 
sion of the colors can be obtained by means of some device 
like that described in the following experiment. If sunlight 
is not available, the stereopticon, if provided with a calcium" 
light or an electric arc-light, can be successfully used, the 
prism being placed in the path of the rays after they have 
traversed the projecting lens. 

EXPERIMENT. 

By means of a porte-lumiere (No. XXX ) throw a beam of sunlight 
through a narrow slit at S, Fig. 116. Place a lens, L, in the path of 
the beam, and adjust it so as to throw a distinct image of the slit 
on a screen at I. Now introduce a prism, P (2s o. XXXII), in the 



OPTICAL INSTRUMENTS. 159 

position shown in the figure, and then place the screen at RR', 
making the distance PR equal to PI The prism used may be of 
flint-glass, or, better, may be hollow and filled with the highly dis- 
persive liquid bisulphide of carbon. 

Examine the spot of colored light on the screen. (1) How many 
colors can be distinctly seen ? (2) Do they blend, or are they sharply 



Fig. 116. 

separated from each other? (3) Which color is most refracted? 
least refracted? Try the effect of passing the emergent pencil 
through a second prism similar to the first, and placed so as to re- 
fract the light in the same direction as the first. 

Try the effect with a second prism so placed as to refract in the 
opposite direction from the first. 

163. The Simple Microscope. — In its least complicated 
form the simple microscope, or magnify] ng-glass, consists 
of a convex lens used, as explained in § 145, to form an 
upright magnified image of any small object. When much 
magnifying-power is required, two or even three convex 
lenses, mounted one over the other with their surfaces only 
a few millimeters apart, are often used. Such combinations 
are called doublets or triplets, according to the number of 
lenses composing them. They have certain advantages over 
single lenses of equal magnifying-power. 

The discussion in § 145 will helj) the student to see that 
the magnifying-power of a simple microscope is greater as 
its focal length is less. 

164. The Compound Microscope, — For viewing objects 
under any but the lowest magnifying-powers, that is, in all 



160 



PHYSICS. 



cases wher^ the apparent diameter of the image is to be 
anywhere from 50 to 5000 times the actual diameter of the 
object, the compound microscope is employed. The essen- 
tial optical parts of this instrument, as usually constructed, 
are (see Fig. 117), an eye-piece, LL\ here represented as 




single, but generally consisting of two convex lenses, and 
an objective, 11, frequently consisting of from two to six 
pieces. These lenses are fixed in a brass tube so arranged 
that the distance between the eye-piece and the objective 
can be varied at will, within certain limits. A mirror, not 
here shown, which is adjustable to any desired angle, is 
asually employed for throwing light upon the object. 
The object to be viewed is placed on a platform beneath 



OPTICAL INSTRUMENTS. 161 

the objective, and is strongly illuminated by light reflected 
from the mirror. A real, inverted, magnified image, A x B l9 
of the object is formed within the tube of the instrument 
at a position somewhat nearer to the eye-piece than its 
principal focus. This real image is therefore magnified by 
the eye-piece, which forms an enlarged virtual image, A' B\ 
of it at a position not far from the object. 

The foci of the object-glass are at/ and/', those of the 
eye-piece at F' and F. 

The total magnifying -power of the instrument is that of 
the objective multiplied by that of the eye-piece. In 
general, the shorter the focal length (see Appendix I) of a 
microscope objective, the greater its magnifying-power. 

An objective of one inch focal length will, on a tube 10 
inches long, give, with the lowest power eye-piece in com- 
mon use (the " A " eye-piece), a magnification of about 50 
diameters; with an eye-piece of double the magnifying- 
power (" B " eye-piece) the total magnification will be 
about 100 diameters, and so on. 

EXPERIMENT. 

Fasten a page of fine print, P in Fig. 118, upright on a table in a 
good light. Set up in front of it a short-focus convex lens, L, at a 
distance from the page somewhat greater than the focal length. 



If 



Fig. 118. 

Hold another short-focus convex lens, L\ in various positions farther 
from the page until one position is found in which an eye close to L' 
sees through it an inverted, magnified image of the print, this being 
a virtual image of the real image formed by the lens L. This appa- 
ratus is a rude model of the compound microscope. 



162 



PHYSIOS. 



165. The Astronomical Refracting Telescope. — This in- 
strument consists essentially of the long-focns object-glass, 
or objective, L (Fig. 119), mounted in one end of a tube, at 




Fig. 119. 

the other end of which is placed an eye-piece, L\ precisely 
similar to that of the compound microscope. The eye- 
piece can be moved toward or away from the object-glass 
in order to make the image appear most distinct. 

The real image of any distant object is, of course, always 
formed by the objective very near its principal focus. The 
foci of the eye-piece are at i^and F'. 

Astronomical telescopes are always furnished with achro- 
matic object-glasses (§ 149). 

EXPERIMENT. 

Mount upon blocks two convex lenses, one of 30 or 40 cm. focal 
length, the other of about o cm. focal length. Set them up on the 
table with their principal axes coincident— that is, with their centres 
on the same straight line at right angles to the centres of their faces. 
Mount a bit of tracing-paper or greased writing-paper, and place 
this screen in such a position between the lenses that the one of 
greatest focal length shall throw upon it a distinct image of some 
distant bright object. Look at this image on the translucent paper 
through the 5-cm. lens Choose such a position and distance as to 
give a clea- virtual image, as much magnified as possible, of the 
real image on the screen. Now remove the screen, and observe that 
the virtual image of the real image is still visible. 

166. Efficiency of the Telescope. — The usefulness of the 
telescope as an aid to vision depends upon the following 



QUESTIONS AXD PROBLEMS. 163 

points: (a) the clearness and sharpness of the image, or 
what is called the definition of the instrument; (b) the 
brilliancy of the image; (c) the amount of allowable mag- 
nification. 

Good definition depends upon the accuracy with which 
the leas is shaped and finished, and upon the quality of the 
glass, which should be free from flaws. 

Brightness depends upon the amount of light which can 
be concentrated in the different parts of the image. Hence 
a large objective will, other things being equal, give the 
best illumination. In some recent telescopes the objective 
has a diameter of 3 feet or more. 

The magnification, with a given eye-piece, is evidently 
very nearly proportional to the focal length of the objec- 
tive; but unless the objective is large, and furnishes much 
light, it is useless to give it great focal length, for the 
reason that the much-magnified image would be too faint 
to be seen to advantage. 

QUESTIONS AND PROBLEMS. 

(1) How could you find the weight of a body that will float, if you 
had no balance but had a vessel filled with water and a "graduated * 
glass flask — that is, a flask with marks upon it showing the number 
of cu. cm. required to fill it to certain depths ? 

(2) If a liter of hydrogen weighs .0896 gin. and if the sp. gr. of 
oxygen as compared with hydrogen is 16, what is the weight of 1 
cu. m. of oxygen ? 

(3) A certain volume of mercury of density 13.6 weighs 216 gm., 
and the same volume of another liquid weighs 14.8 gin. Find the 
density of the second liquid. 

(4) A piece of iron weighs 200 lbs. in air and 172.5 lbs. in water. 
How great is its sp. gr. ? 

(5) A given body weighs 500 gm. in air and 400 gm. in water. 

(a) How great is its volume ? 

(b) How great is its sp. gr. ? 

(6) A board 12 X 6 X 1 in. weighs 1.5 lbs. What is its density in 
lbs. per cu. ft.? 



164 PHYSICS. 

(7) A cubical block of wood 15 cm. along the edge weighs 1125 
gm. What is its density ? 

(8) A 30 cu. cm. body weighs 10 gm. in water. How great is its 
sp. gr. ? 

(9) What is the volume of a body which weighs 25 gm. in air and 
20 gm. in water ? 

(10) A body weighs 180 lbs. in water and 120 lbs. in a liquid that 
is 1.8 times as dense as water. Find the volume and the sp. gr. of 
the body ? 

(11) How much will a kgm. weight of sp. gr. 7 weigh in a liquid 
which is 0.8 as dense as water ? 

(12) A cubical box, 3 ft. square on a side, made of 2 in. plank of 
sp. gr. 0.5, has a bottom but no top. It contains a body weighing 
100 lbs. To what depth will this box sink, upright, in water ? 

Ans. 6.7 in. nearly. 

(13) The sp. gr. of air, as compared with water, is about .00129 at 
0° C. under ordinary atmospheric pressure. How many grams 
would equal the buoyant force exerted by air in this condition upon 
a cu. m. of any substance? 

(14) If the sp. gr. of a certain block is 0.3 and its volume 100 cu. 
cm. , how much of it would be submerged if it were floating in a 
liquid of sp. gr. 2. 

(15) A rod floats one-half submerged in a liquid of sp. gr. 0.9. 
How much of it would be submerged in a liquid of sp. gr. 3? 

(16) There is a uniform rod 6 ft. long and 4 in. square, of sp. gr. 
0.5. What must be the sp. gr. of a cubical piece of metal 4 in. on 
the edge which, when attached to the rod, would just hold it sub- 
merged in water ? 

(17) If a diver with his suit weighs 200 lbs. and it takes ^ of a 
cu. ft. of lead, sp. gr. 11.4, to keep him submerged in fresh water, 
how many cu. ft. of water does he, in his suit, displace? 

(18) Two boys are pulling at a rope in opposite directions, each 
with a force of 25 lbs. 

(a) How great is the tension on the rope? 

(b) How great would you call the tension if the rope were tied to a 
beam and supported a weight of 25 lbs. ? 

(19) A uniform beam, 12 ft. long and weighing 300 lbs., rests, 
horizontal, on a fulcrum 2 ft. from one end. How much weight 
must be applied at this end to make the beam balance in its present 
position ? 

(20) (a) Find the direction, position, and magnitude of the equil- 



QUESTIONS AND PROBLEMS. 165 

ibrant (§ 74) of two forces, parallel and in the same direction, one of 
which is 10 lbs. and the other 12 lbs., their lines of action being 3 ft. 
apart. 

(b) Find the direction, position, and magnitude of the resultant 
(§ 75) of the same two forces. 

(21) One end of a horizontal beam 20 ft. long and weighing 50 lbs. 
rests upon a wall, and the other end is supported by a rope that will 
bear only 85 lbs. A boy weighing 100 lbs. walks slowly along the 
beam from the wall toward the rope. How far from the rope will 
the boy be when it breaks ? 

(22) A hammer is use! to draw out a nail from a board. The 
head of the hammer rests against the board at a distance of 3 in. 
from the nail. A force of 50 lbs. is applied at right angles with the 
handle at a point 12 inches from the boar^l. How great is the force 
exerted by the hammer on the nail ? (This case is similar in principle 
to some of those discussed in connection with the pulley. See Ex- 
periments under § 58.) 

(23) If a force of 50 lbs. is applied at the end of the handle of a 
"jack-screw " 18 in. from the centre of the screw, and if one revolu- 
tion of this screw lifts a weight 0.5 in., how great is this weight, 
if there is no frict.on ? 

(24) A sled weighing with its load 50 lbs. rests on the side of a 
hill rising 1 ft. in a distance of 5 ft. along the incline. 

(a) How great a force acting parallel to the incline is needed to 
keep the sled from sliding downward if there is no friction ? 

(b) If the crust on the snow is just strong enough to bear the sled 
under these conditions, how much would the load on the sled have 
to be lightened in order that a similar crust might bear the sled on a 
level ? 

(25) An inclined plane rising at an angle of 45° has a load of 50 
lbs. resting upon it. How large a horizontal force will be needed to 
keep this load moving up the incline if there is no friction ? 

(26) A horizontal force of 10 lbs. is required to keep a certain body 
moving along a horizontal surface with which its coefficient of fric- 
tion is 0.2. How great is the weight of the body ? 

(27) A mass of 100 lbs. rests upon an inclined plane 10 ft. long and 
4 ft. high. 

(a) How great must be the resistance of friction to keep the body 
from sliding down the incline ? 

(b) How great must the coefficient of friction be ? 

(28) If a simple pendulum 1 m. long vibrates 58 times a minute, 



166 PHYSICS. 

what is the length, of a simple pendulum that vibrates 116 times in 
a second ? 

(29) The length of a simple pendulum vibrating once a f econd in 
the latitude of New York is about 39.1 in. How many seconds a 
day would a clock lose if controlled by a simple pendu'um 40 in. long ? 

(30) Two lights, A and B, are placed 20 ft. apart. The power of A 
is to that of B as 4 to 9. At what point between them must a screen 
be placed in order to be equally lighted up on both sides ? 

(31) The distance of the planet Neptune from the sun being 
2,800,000,000 miles, nearly, how long does it take a wave of light to 
go from the sun to Neptune ? 

(32) What is the height of a tree which casts a shadow 100 ft. long, 
when an upright rod 5 ft. tall casts a shadow 7 ft. long ? 

(33) The image of an upright stake 8 ft. tall, and 10 ft. from a 
window-shutter appears on a screen 4 ft. beyond the shutter. The 
aperture in the shutter through which the light passes from the stake 
to the screen is very small. How great is the length of the image ? 

(34) The clock on a wall indicates 9.30. What time will it appear 
to indicate if the observer sees the reflection of the clock in a 
mirror on the opposite wall but does not distinguish the numerals ? 

(35) A plane mirror lies up m a table and a pencil 6 in. tall stands 
upright on one edge of the mirror. How wide must the mirror be 
in order that a person whose eyes are 5 in. above its surface and 20 
in. distant from the pencil may just see the whole length of the 
pencil reflected in the mirror ? (To be solved by drawing and meas- 
uring. The thickness of the glass is to be neglected. ) 

(36) Prove that if an object is placed in front of a plane mirror and 
the mirror is moved either toward or from the object, without turning, 
the image will move twice as far as the mirror. 

(37) Prove that if a candle is placed in front of a vertical plane 
mirror and the mirror is turned 45° about a vertical axis, the image 
of the candle will move through an arc of 90° around the axis of 
the mirror. 

(38) Two plane mirrors, A and B, are placed 12 cm. apart, facing 
each other and parallel. A small object is placed between them 4 cm. 
distant from A. Calculate the distance from A to the first and 
second images seen in it. Do the same for B. 

(39) Two plane mirrors, placed vertical, make with each other an 
angle of 60°. A candle is placed between them, but nearer one than 
the other. Draw a figure showing the positions of the various images 
of the candle. 



QUESTIONS AND PROBLEMS. 167 

(40) If the radius of curvature of a concave spherical mirror is 50 
cm., and if a candle is placed 40 cm. distant from the mirror, 

(a) How far from the mirror will the image of the candle be ? 

(b) Will this image be real or virtual ? 

(c) Will it be erect or inverted ? 

(d) If the candle-flame is 2 cm. long, what will be the length of its 
image ? 

(41) If the candle mentioned in the preceding problem was 10 cm. 
from the mirror, what would be the answers to the questions there 
stated ? 

(42) What would be the answers in problems 40 and 41 if the 
mirror were convex ? 

(43) Have you ever seen curved mirrors used except in a class-room 
or laboratory? If so, for what purposes were they used? 

(44) Define the term index of refraction. 

(45) The index of refraction of the earth's atmosphere is little greater 
than 1 with respect to the space outside this atmosphere. Does this 
fact delay, or does it hasten, the first glimpse of the rising sun ? 

(46) For which of the colors here named is the index of refraction 
of glass the greatest — red, green, yellow, blue ? For which of them 
is it least ? 

(47) How could you find by experiment the color complementary to 
any given tint ? 

(48) Show that the image formed by a convex lens may be either 
larger or smaller than the object. 

(49) Prove algebraically, and also graphically (after the manner of 
§ 142), that when thy distance of an object from a convex lens is 
twice the focal length, the image is at the same distance on the other 
side. 

(50) A rod 5 cm. long held in front of a convex lens, at right angles 
with the principal axis, has an image 25 cm. long upon a screen dis- 
tant 100 cm. from the lens. How great is the focal length of the 
lens? 

(51) An object 4 cm. long, placed 20 cm. from a certain lens and 
at right angles with the principal axis, has a real image 10 cm. dis- 
tant from the lens. If the same object were placed 5 cm. distant 
from the same lens, 

(a) Would the image be real or virtual ? 

(b) How far from the lens would the image be ? 

(c) How great would the length of the image be ? 



APPENDIX I. 

FOCAL LENGTH, ETC., OF LENSES AND COMBINATIONS OF 

LENSES. 

It is customary to define the focal length, F y of a single lens as the 
distance from the focus to the nearest point of the surface of the lens, 

and in the formula — = — - -f- — to consider D and Di as measured 
Jo JJo JJ\ 

from the object and image, respectively, to the nearest point of the 
lens. With this interpretation of the letters, the formula is not ex- 
actly fulfilled by any actual lens. It holds strictly true only for the 
ideal case of a lens of zero thickness, but it is sufficiently near the 
truth for common purposes in the case of ordinary lenses. The for- 
mula is about equally accurate, for a4ouble convex lens, at least, when 
all the distances, F, D , and Di, are measured to the optical centre of 
the lens (§ 137). 

When a combination of lenses is used, as in a microscope-objective 
or a photographic camera, a formula similar to that just given can be 
applied, but the F, D , and Di occurring in it are not now measured 
either to the nearest point of the combination or to the optical centre. 
They are measured to certain other points determined by the radii of 
curvature, thickness, and refractive index of each lens, and the dis- 
tance between the two lenses. In the ordinary use of such a combi- 
nation, its magnifying power is substantially equivalent to that of a 
single ideal thin lens having a focal length equal to what is called 
the focal length of the combination. The calculation of the focal 
length of the combination is frequently very laborious. 

Dealers in photogiaphic objectives very frequently state as the 
focal length of a combination of lenses the distance from the principal 
focus to the near, r surface of the nearest lens. They sometimes call 
this the " back focal length,' ' or, rather, the * * back focus" of the coni- 

168 



APPENDIX I. 



169 



bination. It is a convenient quantity to use in the description of a lens, 

but it is not intended for use in the formula -= — — 4- -=r . 

F D G Z>i 
The term " equivalent focal length/' or " equivalent focus," is 
sometimes applied, in the case of a combination of two equal lenses, 
to the distance from the principal focus to a point midway be- 
tween the two lenses. 



APPENDIX II. 

INDICES OF REFRACTION OF VARIOUS SUBSTANCES 
COMPARED WITH A VACUUM. (See § 128.) 



Agate 1.540 

Canada balsam 1.53 

Diamond 2.5 

Fluor spar 1.434 

Glass (ordinary crown). 1.53 
" ( " flint)... 1.61* 

Ice 1.31 

Quartz 1.544 

Rock salt 1.544 



Selenium (crystals) 



2.98 



Alcohol .., 1.36 

Petroleum (heavy) 1.45 

(light) 1.4 + 

Water 1.333 

Nitrogen 1.000298 

Oxygen 1.000371 



* The dispersive power (§ 134) of flint glass is nearly twice as great 
as that of crown glass. 



APPENDIX III. 

All the articles in the first list here given should be furnished to 
each member of the laboratory section. 

LIST OF ARTICLES REFERRED TO BY NUMBER IN THE 
"EXERCISES" OF THIS BOOK. 

No. 1. A 10-cni. section of a meter-rod. 

No. 2. A meter-rod, marked on one side in feet and inches. 

No. 3. A 30- cm. bevel- edged measuring-stick, marked on one side 
in inches. 

No. 4. A waterproofed wooden cylinder about 8 cm. long and 4.5 
cm. in diameter, loaded internally with shot so that it will float 
nearly submerged in water. 

No. 5. A brass can about 14 cm. tall and 7 cm. in diameter, having 
a slightly declining, straight, overflow-tube, about 6 cm. long and 
0.8 cm. in internal diameter, extending from a point about 1.5 cm., 
clear, below the top of the can (see Fig. 6). To prevent dribbling 
the junction of tube and can should be covered, internalJy, with a 
coat of paraffin melted on. 

No. 6. A braes catch-bucket with a wire handle, capable of holding 
about 175 gm. of water, and weighing not more than 50 gm. 

No. 7. An 8-oz. spring-balance graduated to 0.5 oz. (There is now 
in the market an improved balance, graduated on one side in 10-gm. 





[Tni^iiiliinfiinjiinf 1 

Fig. 120. 

intervals and on the other side in 0.25-oz. intervals. It is, moreover, 
especially adapted for use in the horizontal position. This improved 
balance is desirable for this course.) 

170 



APPENDIX III 171 

No 8. A rectangular waterproofed block of wood, about 7 cm. 
long and 4.5 cm. square on the end, so loaded internally with shot 
that it will sink in water, but not enough to make it weigh more 
than 225 gm. 

No. 9. A rectangular waterproofed cherry block about 7.5 cm. X 7.5 
cm. X 3.8 cm. This block should be smooth, and therefore the water- 
pr ioflng should be done by soaking it in very hot paraffin. For the 
best results this soaking should be done in a vacuum. Excess of 
paraffin should be scraped off before the block is used. 

No. 10. A one-gallon glass jar of good quality. (It is poor economy 
to buy a poor jar and have it break with a liquid in it.) 

No. 11. A lump of roll sulphur weighing about 175 or 200 gm. 
It is not worth while to cast these lumps into regular cylindrical 
form. 

No. 12. A lead sinker with w r ire handle, weighing about 175 gm. 

No. 13. A waterproofed wooden cylinder about 1 cm. in diameter 
and 20 cm. long. Doweling-rod, furnished by hardware dealers, 
serves well when waterproofed. 

No. 14. A holder for keeping No. 13 upright in water. It consists 
of a waterproofed wooden rod about 12 cm. long and 1.3 cm. square 
on the end, provided with a clasp for attaching it to the side of a jar, 
and with two screw-eyes projecting from one side, the rings of which 
are large enough to let the cylinder No. 13 slip easily through them, 
but not large enough to allow the cylinder to tip far from the vertical 
position (see Fig. 10). 

No. 15. A cylindrical glass jar, about 14 cm. tall and 10 cm. in 
diameter, with level top. 

No. 16. A broad-mouthed bottle with ground-glass stopper, stand- 
ing not much more than 11 cm. tall with stopper, and weighing, 
when filled with water, about 175 or 200 gm. 

No. 17. A lever and supporting-bar. The lever is a 30-cm. section 
from a meter-rod, pivoted upon the smoothed cylindrical body of a 
brass screw which is driven horizontally into the end of a bar of 
hard wood about 25 cm. long, 5 cm. wide, and 3 cm. thick. A 
brass plate projecting from this bar and overhanging the middle of 
the lever prevents the lever from tipping far, while it allows suffi- 
cient freedom of motion. The lever itself, except for a distance of 
2 cm. each side of the middle, is cut away so that its top is level with 
the upper part of the hole through the centre. There should be a 



172 PHYSICS. 

screw-hole running downward through, the middle of the supporting- 
bar, to facilitate in attaching it, as shown in Fig. 21. 

No. 18 (A and B). Two brass scale-pans about 6.5 cm. square, 
each with its suspending threads weighing accurately 1 oz. (that is, 
not differing from this weight by more than .01 oz.). Each pan is 
suspended by four strong linen threads meeting in a knot about 20 
cm. above the pan, two of them continuing in a loop about 4 cm. 
long above this knot. (Fig. 21.) 

No. 19. A set of iron weights, 8 oz , 4 oz., 2 oz., and two 1 oz. , 
making a total of 16 oz. No weight should be in error more than 
.01 oz. 

No. 20. A flat pine board about 50 cm. long and 15 cm. wide for 
use in the Exercises on Friction. 

No. 21. A cubical block of wood about 3.7 cm. on each edge. A 
groove about 1 cm. wide and 2 cm. deep extends through the lower 
part of the block with the grain of the wood. An ordinary short 
screw extends through one side of the block into this groove, and 
serves to fix the block in position upon a meter- rod. Across the grain 
at the top of the block is a slot about 0.1 cm. wide and 0.5 cm. deep. 
(Figs. 26 and 105.) 

No. 22. Two bits of wood, each about 8 cm. long and 1 cm. square 
on the end, for supporting the spring- balance in a horizontal position. 
(Fig. 41.) 

No. 23. A plate-glass mirror about 15 cm. long, 3.8 cm. wide, and 
0.2 cm. thick, the coating on the back protected by paint or varnish. 

No. 24 (A and B). Two straight- edged rulers of some wood that 
will keep its shape well — white pine, for instance — each about 30 
cm. long, 5 cm. wide, and 1 cm. thick. 

No. 25. A block like No. 21, but without the large slot and the 
screw. One side of this block is coated with white paper, and a 
vertical pencil-mark or ink-mark is made across the middle of this 
paper. (Fig. 108.) 

No. 26. A Walter Smith " school {rquare," or other equally good 
protractor. 

No. 27. A cylindrical mirror of nickel-plated brass, about 5 cm. 
tall and 8 cm. wide, cut from seamless tubing 4 inches in diameter 
and J inch thick, mounted upon a sem circular base-board of wood 
of the proper radius of curvature. The base-board should be about 
1.5 cm. thick. 

No. 28. A piece of plate-glass about 7 cm. square and 0.6 cm. thick, 



APPENDIX III. 178 

for Exercise 20 on Index of Refraction. Two opposite edges or narrow 
sides of the glass should be ground tolerably plane and polished 
sufficiently to allow seeing readily through the whole width of the 
plate (see Fig. 89). 

No. 29. A brass partition made to fit the small glass jar (No. 15) 
and to extend downward into the jar a distance equal to about one- 
third the diameter of the jar. It should be made of sheet brass 



Fig. 121. 

about .07 cm. thick, The method of shaping and adjusting the par- 
tition is suggested by Fig. 121, where A shows a side view and B an 
end view of the partition. The flanges shown in B are bent more or 
less in adjusting the partition to fit the jar closely, but without too 
much pressure. 

No. 30. An index of thin sheet brass made to clasp the side of the 
jar (No. 15). This index is a strip about 15 cm. long, before bending, 
and 1 cm. wide, tapered to a point at one end. To enable it to clasp 
the jar, about 3 cm. at the untapered end is bent over. (See pb in 
Fig. 91.) 

No. 31. A circular (not elliptical) double-convex spectacle-lens, 
having a focal length not less than 12 cm. and not more than 16 cm. 
The lens is mounted on a block similar to No. 21. (See Fig. 105.) 

No. 32. A white cardboard screen about 8 cm. square, of such 
thickness as to be held firmly in the narrow slot of the small block 
No. 21. (Fig. 105.) 

No. 33. A small kerosene lamp of such size and shape as to fit it 
for the use shown in Fig. 106. The lower partof the chimney is sur- 
rounded by a thin sheet of asbestos paper, having a hole 3 or 4 mm. 
in diameter at the height of the flame. 

No. 34. A wire, of the right size to fit into the narrow slot of No. 
21, bent at a right angle, one arm about 6 cm. long, the other about 
4 cm, (Fig. 108.) 



174 



PHYSICS. 



ARTICLES USED BY THE TEACHER, BUT NOT TO BE 
FURNISHED TO STUDENTS. 

(Most of them are referred to by number in the " Experiments " of 

this book.) 

No. I. A gauge for testing pressure at various points and in vari- 
ous directions in a jar of water. In Fig. 122, P is a pillar of wood or 
metal about 25 cm. tall ; G is a small glass thistle-tube about 1.7 cm. 
wide ; m is a tbin rubber membrane fastened water-tigbt across the 
mouth of C ; p and p are hard-rubber pulleys about 1.7 cm. in diam- 




eter, fitting closely on their axes ; r is a small rubber tube ; g is a 
glass tube ; i is a short column of water serving as an index. A band 
of strip-rubber, such as toy stores supply, connects the two pulleys p 
and p, so that by turning a, the axis of the upper pulley, between the 
thumb and finger, the gauge-face m may be turned upward, down- 
ward, or sidewise, without changing level. A student-lamp chimney, 
with stopper for one end, accompanies this gauge. 



APPENDIX III 175 

No. II, Apparatus for bursting a bottle by an attempt to compress 
water within it. 

The essentials are a glass bottle, with a perforated rubber stopper 
which fits the bottle well when driven in its full length ; a strong 
frame for holding the bottle and keeping the stopper in place ; a rod, 
with convenient handle, to be driven water-tight down through the 
hole in the stopper, 

No. III. Glass tube about 1 m. long, closed at one end, connected 
by a strong rubber tube 25 cm. long with another glass tube 20 cm. 
long. .See Fig. 12.) 

No. IV. Strong thistle-tube (Fig. 13) about 2.5 cm. wide, covered at 
the mouth with strong sheet rubber and furnished with a thick- 
walled rubber tube about 20 cm. long. 

No. V. Small air-pump suitable for both exhaustion and compres- 
sion. 

Such a pump is frequently sold without base, but it is well to have 
a base, bell-jar plate, and one or two bell-jars. For many purposes a 
larger pump is desirable. 

No. VI. Bent glass tube for Boyle's law, the whole tube about 1.5 m. 
long (Fig. 14). 

No. VII. Common large rubber foot-ball, with a rubber tube about 
30 cm. long attached to the key. (Fig. 15.) 

No. VIII. Small bottle provided with rubber stopper fitted with 
two glass tubes as in Fig. 16. 

No. IX. Glass model of lifting-pump (Fig. 17). 

No. X Glass model of force-pump (Fig. 18). 

No. XI. Hydrometer for liquids less dense than water. 

No. XII. Hydrometer for liquids more dense than water. 

No. XIII. Glass U tube (Fig. 20) about 60 cm. long before bending. 

No. XIV. Some form of the Cartesian Diver. 

No. XV. Eight-inch and four-inch wooden disks combined in one 
piece for use as a pulley. This piece is fitted with various pins (re- 
movable) for suspending weights. It is mounted much like the lever 
of No. 17. (See Fig. 34). 

No. XVI. Centre-of-gravity board, with suspension and plummet. 
(Fig. 24.) 

No. XVII. Platform balance weighing from 1 kgm. to 0.1 gm., 
provided with a set of brass weights. 

No. XVIII. Well-made small brass pulley with a hook or loop. 
(Fig. 37.) 



1Y6 



PHYSICS. 



No. XIX. Well-made small double brass pulley with hook or loop. 
(Fig. 38.) 

No. XX. An inclined plane,* shown about one-fourth natural size 
in Fig. 123. The roller should be of brass, accurately turned. It 
weighs with its frame jus<- 16 oz. The graduations of the scale may 
be in millimeters. The apparatus should be made with care. 

No. XXI. Pendulum-support and pendulum-balls (Figs. 56 and 57). 




Fig. 123. 



No. XXII. Three small packages of dyestuffs soluble in water, 
various colors. 

No. XXIII. Three glass plates, red, green, and blue, about 10 cm. 
square. 

No. XXIV. Camera obscura consisting of two pasteboard tubes 
each about 25 cm. long. The larger, about 5 cm. in diameter, is 
closed at one end save at the centre, where there is a hole about 0.1 
cm. in diameter in a thin partition. The smaller tube, about 4 cm. 
in diameter, is closed at one end by thin tracing-paper. (See § 94.) 

* A number of excellent features in this apparatus are due to Mr. 
Sweet, formerly of the Rindge Manual-Training School in Cambridge. 



APPEXDIX II J 177 

No. XXV. Make according to the following directions : On a board 
about 35 cm. square (Fig. 73) lay off a circle 30 cm. in diameter. 
Bore 12 holes, 1, 2, 3, etc., dividing the circumference in 30° parts. 
From the centre draw radii, making the angle a of 90°, /3 of 60°, and 
y of 30°. Provide pegs, about 15 cm. tall, to fit in all the holes. 

Mount two strips of thin " silvered '' glass, each about 20 cm. long 
and 10 cm. wide, on two boards hinged together in such a way that 
the angle between them may be varied from 30° or less to 90 D or 
more, the longer edges of the mirrors being horizontal. 

No. XXVI. An inexpensive kaleidoscope. 

No. XXVII A concave spherical mirror 12 or 15 cm. in diameter. 

No. XXVIII. A " granite-ware " ba^in 15 cm. or more in diameter. 

No. XXIX. Thin waterproofed board, pierced by knitting-needle 
for experiment on the critical angle. (Fig. 96.) 

No. XXX. A porte-lumiere. 

No. XXXI. Right-angled glass prism about 5 cm. long, for show- 
ing "total reflection." (Fig. 95.) 

No. XXXII. A pair of equilateral prism?, for experiment on pro- 
jecting the solar spectrum. (§ 162.) 

No. XXXIII. Set of about half a dozen lenses of various shapes 4 
or 5 cm. in diameter. 'Fig. 101.) 

No. XXXIV. Set of about half a dozen convex lenses varying from 
2 cm. to 50 cm. in focal length, the largest 6 or 8 cm. in diameter. 

No. XXXV. Rotating apparatus* suitable for carrying Maxwell's 
color-disks, etc. 

No. XXXVI. Set of color-disks, e.g. , those made by Milton Bradley. 

MISCELLANEOUS ARTICLES. 

Two pounds of clean mercury. 

Two pounds of assorted soft glass tubing, from 2 mm. to 8 mm. 
inside diameter. 

Six feet of rubber tubing, about 5 mm. inside, that will not col- 
lapse when connected with the air- p am p. 

An ounce or two of very small rubber tubing. 

Piece of thin sheet rubber about 6 in. square, for use with the 
gauge. (No. I.) 

Set of cork- borers. 

Three-cornered file for cutting glass tubing. 

* The well-known little tops with color-disks serve very well if 
larger forms of XXXV and XXXVI are not available. 



178 PHYSICS. 

Screw-driver. 

Pair of wire-cutting pliers. 

One- half pound of naked copper- wire about 1 rnm. in diameter. 

LABORATORY TABLES. 

The laboratory tables used in the Cambridge grammar-schools are 
well suited to the work of this course. They are about 10 ft. long, 
4 ft. wide, and 2 ft. 10 in. tall. They have white- pine tops about 1^ 
in. thick, and heavy white- wood legs. Extending from end to end 
over each table are two horizontal bars, about 2 in. by 3 in., ad- 
justable at various heights (which should range from 1| ft. to 3£ 
ft. by 3-iu. intervals) above the table-top, their ends, which are 
cut in tenons, sliding in grooves in the supporting posts. These 
posts are fastened to the frame of the table and rise through slots in 
the table-top, being flush with the ends of this top and about 10 
in. distant from the sides. Pins of iron or wood placed in holes 
in these posts support the ends of the horizontal bars. To adapt 
these tables to Exercises 29 and 30 (Second Part of book), holes 
about 1-J in. in diameter should be bored through the top. 

For Exercises in heat (see Second Part) this table should have a 
gas-pipe running along the middle of the top from end to end, with 
three stop-cocks leading to the right and three leading to the left. 
This pipe should be readily detachable, as its pre c ence would be in- 
convenient in many experiments. 

Each table is intended to accommodate six independent experi- 
menters. 



APPENDIX IV. 

A FEW EQUIVALENTS IN THE ENGLISH AND METRIC 
SYSTEMS. 

1 meter = 1.0936 yards. 
1 " = 3.2809 feet. 
1 " = 39.3705 inches. 
1 kilometer = 0.6214 mile. 

1 gram = 15.4323 grains = 0.0353 ounce- 
1 kilogram = 2.2046 pounds avoirdupois. 

1 yard = 0. 9144 meter. 
1 foot = 0.3048 " 
1 inch = 0.0254 " 
1 mile = 1.6093 kilometers. 
1 pound avoirdupois = 0.4536 kilogram. 
1 ounce = 28.35 grams. 

The following are approximate equivale 

1 decimeter = 4 inches. 

1 meter = 1.1 yards. 

1 kilometer = f of a mile. 

1 kilogram = 24 pounds. 

179 



INDEX. 



Aberration, chromatic, spherical, 
149 

Achromatic, prism, 135 ; lens, 
150 

Angle, of incidence and of re- 
flection, 107; of refraction, 125; 
critical, 131 ; refracting, of 
prisms, 134 

Archimedes, principle of, 19 

Atmospheric pressure, 31 

Axis of lens, 137 

Balance, spring-, errors of, 61 
Barometer, Aneroid, 34; Torri- 

celli's 31 
Boyle's law, 35 
Bunsen's photometer, 98 

Camera, obscura, 93 ; photog- 
rapher's, 156 

Centre, of curvature, 112; optical, 
137 

Chromatic aberration, 149 

Coefficient of friction, 80 

Colors, 92, 100 ; complementary, 
154 ; mixing, 153 

Composition of forces, 76 

Conjugate foci, 140 

Critical angle, 131 

Curvature, centre of, in mirror, 
112; in lens, 137 

Density, definition of, 15 ; of 

water, 16 
Dispersion, 135 
Distance, measurement of, 3, 4 ; 

object- and image-, 140 

English and metric units, App V 
Equilibrant, definition of, 70 



Eye, 152 
Eyepieces, 161 

Floating bodies, specific gravity 

of, 23 
Fluids, definition of, 28 ; liquid 

and gas, 34 
Focal length, of mirrors, 115 ; of 

lenses, 138 ; formula for, 142, 

148, 168 
Foci, conjugate, 41 
Focus, principal, of mirrors, 115, 

118 ; of lens, 138 
Foot, relation to meter, App. V 
Forces, parallelogram of, 61-69 
Friction, 78 ; between solids, 80 ; 

coefficient of, 80 ; between 

solids and fluids, 84 ; rolling, 

83 ; in tubes, 84 
Fulcrum, definition of, 49 ; force 

at, 52 

Gas, 28, 34 

(rases, pressure of, 31, 35 
Gravity, specific, 18 ; centre of, 
46 

Hydrostatic press, 36 

Illumination, measure of, 96 

Images, in plane mirrors, 104- 
111 ; by convex mirror, 112 ; by 
concave, 117 ; distorted, 119 ; 
of lenses, 143-148 ; after, 155 

Inch, relation to cm., App. V 

Inclined planes, 69-75 

Incident angle 107 

Index of refraction, 126 

Inverse square, law of, 95 

Kaleidoscope, 111 



181 



182 



INDEX. 



Lantern, magic, 157 

Length, focal, of mirrors, 115 ; 
of lenses, 138 ; formula for, 
142, 148, 168 

Lenses, definitions relating to, 
136-138; shapes of, 136; achro- 
matic, 150 

Lever, definition of, 41 ; circular, 
44; weight of, 46; laws of, 50, 
51, 53, 58 ; pulley, windlass, 
capstan, 56 

Light, 90 ; velocity of, 90 ; theory 
of, 91 ; colors, 92, 100 ; pencils 
and rays, 92 ; weakens with 
distance; 95 

Liquid, 28, 34 

Liquids, pressure of, 28-30, 36 

Machines, 41 

Magic latern, 157 

Mass, 17 

Meter, relation of, to foot, App. 
V 

Metric system, App. V 

Microscope, 159 

Mirrors, plane, 104; cylindrical 
and spherical, 111, 121 ; con- 
vex, 112; concave, 115; for- 
mulas for, 121 



Objectives, 160 
Optical instruments, 



156 



Parallel rays, 140 
Parallelogram, measurement of, 

9, 10; of forces, Chap. V 
Pendulum, 86 
Penumbra, 94 
Photometry, 96 
Physics, definition of, 1 
Plane mirrors, 104 
Porte-lumiere, 160 
Pressure, at different levels, 

liquids, 29, 30 ; gases, 34 ; in 

different directions, liquids, 29; 

gases, 33 
Principal axis of lens, 137 



Prisms, achromatic, 135 ; defini- 
tion of, 133 

Projection, of images and of spec- 
truim. 158 

Pulley, 54-57 

Pumps, 38 

Rays, parallel, 140 

Reflection, of light, 103; from 
plane mirror, 104 — law of, 107, 
114 ; total internal, 131 

Refraction, 125 ; index of, 126; of 
glass, 127, 133; of water, 138; 
of air, 129; of different colors, 
130; relation to velocity, 130 

Resultant of forces, 76 

Rumford photometer, 97 

Screw, 74 

Shadows, 94 

Siphon, 39 

Specific gravities, definition of, 

18; formulas, 20; methods of 

obtaining, 18-27, 39 
Spectrum, 35 

Spring-balance, errors of, 61 
Springs, in watches, 89 
Stereopticon, 157 

Telescope, 162 
Torricelli's barometer, 31 
Triangle, 6 
Tubes, flow of fluids in, 84 

Umbra, 94 

Units of measurement, 5 

Velocity of light, 90 

Virtual image, of mirrors, 108 ; 

of lenses, 146 
Volume, measurement of, 10-13 

Water, density of, 16 
Wave-lengths varying in differ- 
ent colors, 92 
Wedge, 74 
Weight, compared with mass, 17 



